X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=ca3df5fea55949225a51f8255a2c13a33c37d9f2;hb=7f55521ab4652d3ca10cd085f8e9d41bf149e8e5;hp=1250fbda7981aba5474af0a3d2615c969bc9d1e1;hpb=4c8f9aac69d1cb4097b60b10e5b198b6372ec55e;p=sage.d.git

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index 1250fbd..ca3df5f 100644
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -1,9 +1,56 @@
 """
-Euclidean Jordan Algebras. These are formally-real Jordan Algebras;
-specifically those where u^2 + v^2 = 0 implies that u = v = 0. They
-are used in optimization, and have some additional nice methods beyond
-what can be supported in a general Jordan Algebra.
-
+Representations and constructions for Euclidean Jordan algebras.
+
+A Euclidean Jordan algebra is a Jordan algebra that has some
+additional properties:
+
+  1.   It is finite-dimensional.
+  2.   Its scalar field is the real numbers.
+  3a.  An inner product is defined on it, and...
+  3b.  That inner product is compatible with the Jordan product
+       in the sense that `<x*y,z> = <y,x*z>` for all elements
+       `x,y,z` in the algebra.
+
+Every Euclidean Jordan algebra is formally-real: for any two elements
+`x` and `y` in the algebra, `x^{2} + y^{2} = 0` implies that `x = y =
+0`. Conversely, every finite-dimensional formally-real Jordan algebra
+can be made into a Euclidean Jordan algebra with an appropriate choice
+of inner-product.
+
+Formally-real Jordan algebras were originally studied as a framework
+for quantum mechanics. Today, Euclidean Jordan algebras are crucial in
+symmetric cone optimization, since every symmetric cone arises as the
+cone of squares in some Euclidean Jordan algebra.
+
+It is known that every Euclidean Jordan algebra decomposes into an
+orthogonal direct sum (essentially, a Cartesian product) of simple
+algebras, and that moreover, up to Jordan-algebra isomorphism, there
+are only five families of simple algebras. We provide constructions
+for these simple algebras:
+
+  * :class:`BilinearFormEJA`
+  * :class:`RealSymmetricEJA`
+  * :class:`ComplexHermitianEJA`
+  * :class:`QuaternionHermitianEJA`
+  * :class:`OctonionHermitianEJA`
+
+In addition to these, we provide two other example constructions,
+
+  * :class:`JordanSpinEJA`
+  * :class:`HadamardEJA`
+  * :class:`AlbertEJA`
+  * :class:`TrivialEJA`
+
+The Jordan spin algebra is a bilinear form algebra where the bilinear
+form is the identity. The Hadamard EJA is simply a Cartesian product
+of one-dimensional spin algebras. The Albert EJA is simply a special
+case of the :class:`OctonionHermitianEJA` where the matrices are
+three-by-three and the resulting space has dimension 27. And
+last/least, the trivial EJA is exactly what you think it is; it could
+also be obtained by constructing a dimension-zero instance of any of
+the other algebras. Cartesian products of these are also supported
+using the usual ``cartesian_product()`` function; as a result, we
+support (up to isomorphism) all Euclidean Jordan algebras.
 
 SETUP::
 
@@ -13,13 +60,11 @@ EXAMPLES::
 
     sage: random_eja()
     Euclidean Jordan algebra of dimension...
-
 """
 
-from itertools import repeat
-
 from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra
 from sage.categories.magmatic_algebras import MagmaticAlgebras
+from sage.categories.sets_cat import cartesian_product
 from sage.combinat.free_module import CombinatorialFreeModule
 from sage.matrix.constructor import matrix
 from sage.matrix.matrix_space import MatrixSpace
@@ -31,7 +76,7 @@ from sage.rings.all import (ZZ, QQ, AA, QQbar, RR, RLF, CLF,
                             QuadraticField)
 from mjo.eja.eja_element import FiniteDimensionalEJAElement
 from mjo.eja.eja_operator import FiniteDimensionalEJAOperator
-from mjo.eja.eja_utils import _mat2vec
+from mjo.eja.eja_utils import _all2list, _mat2vec
 
 class FiniteDimensionalEJA(CombinatorialFreeModule):
     r"""
@@ -39,16 +84,50 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
 
     INPUT:
 
-      - basis -- a tuple of basis elements in their matrix form.
+      - ``basis`` -- a tuple; a tuple of basis elements in "matrix
+        form," which must be the same form as the arguments to
+        ``jordan_product`` and ``inner_product``. In reality, "matrix
+        form" can be either vectors, matrices, or a Cartesian product
+        (ordered tuple) of vectors or matrices. All of these would
+        ideally be vector spaces in sage with no special-casing
+        needed; but in reality we turn vectors into column-matrices
+        and Cartesian products `(a,b)` into column matrices
+        `(a,b)^{T}` after converting `a` and `b` themselves.
+
+      - ``jordan_product`` -- a function; afunction of two ``basis``
+        elements (in matrix form) that returns their jordan product,
+        also in matrix form; this will be applied to ``basis`` to
+        compute a multiplication table for the algebra.
+
+      - ``inner_product`` -- a function; a function of two ``basis``
+        elements (in matrix form) that returns their inner
+        product. This will be applied to ``basis`` to compute an
+        inner-product table (basically a matrix) for this algebra.
+
+      - ``field`` -- a subfield of the reals (default: ``AA``); the scalar
+        field for the algebra.
+
+      - ``orthonormalize`` -- boolean (default: ``True``); whether or
+        not to orthonormalize the basis. Doing so is expensive and
+        generally rules out using the rationals as your ``field``, but
+        is required for spectral decompositions.
+
+    SETUP::
+
+        sage: from mjo.eja.eja_algebra import random_eja
+
+    TESTS:
 
-      - jordan_product -- function of two elements (in matrix form)
-        that returns their jordan product in this algebra; this will
-        be applied to ``basis`` to compute a multiplication table for
-        the algebra.
+    We should compute that an element subalgebra is associative even
+    if we circumvent the element method::
 
-      - inner_product -- function of two elements (in matrix form) that
-        returns their inner product. This will be applied to ``basis`` to
-        compute an inner-product table (basically a matrix) for this algebra.
+        sage: set_random_seed()
+        sage: J = random_eja(field=QQ,orthonormalize=False)
+        sage: x = J.random_element()
+        sage: A = x.subalgebra_generated_by(orthonormalize=False)
+        sage: basis = tuple(b.superalgebra_element() for b in A.basis())
+        sage: J.subalgebra(basis, orthonormalize=False).is_associative()
+        True
 
     """
     Element = FiniteDimensionalEJAElement
@@ -59,10 +138,13 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
                  inner_product,
                  field=AA,
                  orthonormalize=True,
-                 associative=False,
+                 associative=None,
+                 cartesian_product=False,
                  check_field=True,
                  check_axioms=True,
-                 prefix='e'):
+                 prefix="b"):
+
+        n = len(basis)
 
         if check_field:
             if not field.is_subring(RR):
@@ -71,10 +153,6 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
                 # we've specified a real embedding.
                 raise ValueError("scalar field is not real")
 
-        # If the basis given to us wasn't over the field that it's
-        # supposed to be over, fix that. Or, you know, crash.
-        basis = tuple( b.change_ring(field) for b in basis )
-
         if check_axioms:
             # Check commutativity of the Jordan and inner-products.
             # This has to be done before we build the multiplication
@@ -92,81 +170,74 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
 
 
         category = MagmaticAlgebras(field).FiniteDimensional()
-        category = category.WithBasis().Unital()
+        category = category.WithBasis().Unital().Commutative()
+
+        if associative is None:
+            # We should figure it out. As with check_axioms, we have to do
+            # this without the help of the _jordan_product_is_associative()
+            # method because we need to know the category before we
+            # initialize the algebra.
+            associative = all( jordan_product(jordan_product(bi,bj),bk)
+                               ==
+                               jordan_product(bi,jordan_product(bj,bk))
+                               for bi in basis
+                               for bj in basis
+                               for bk in basis)
+
         if associative:
             # Element subalgebras can take advantage of this.
             category = category.Associative()
+        if cartesian_product:
+            # Use join() here because otherwise we only get the
+            # "Cartesian product of..." and not the things themselves.
+            category = category.join([category,
+                                      category.CartesianProducts()])
 
         # Call the superclass constructor so that we can use its from_vector()
         # method to build our multiplication table.
-        n = len(basis)
-        super().__init__(field,
-                         range(n),
-                         prefix=prefix,
-                         category=category,
-                         bracket=False)
+        CombinatorialFreeModule.__init__(self,
+                                         field,
+                                         range(n),
+                                         prefix=prefix,
+                                         category=category,
+                                         bracket=False)
 
         # Now comes all of the hard work. We'll be constructing an
         # ambient vector space V that our (vectorized) basis lives in,
         # as well as a subspace W of V spanned by those (vectorized)
         # basis elements. The W-coordinates are the coefficients that
-        # we see in things like x = 1*e1 + 2*e2.
+        # we see in things like x = 1*b1 + 2*b2.
         vector_basis = basis
 
-        from sage.structure.element import is_Matrix
-        basis_is_matrices = False
-
         degree = 0
         if n > 0:
-            if is_Matrix(basis[0]):
-                if basis[0].is_square():
-                    # TODO: this ugly is_square() hack works around the problem
-                    # of passing to_matrix()ed vectors in as the basis from a
-                    # subalgebra. They aren't REALLY matrices, at least not of
-                    # the type that we assume here... Ugh.
-                    basis_is_matrices = True
-                    from mjo.eja.eja_utils import _vec2mat
-                    vector_basis = tuple( map(_mat2vec,basis) )
-                    degree = basis[0].nrows()**2
-                else:
-                    # convert from column matrices to vectors, yuck
-                    basis = tuple( map(_mat2vec,basis) )
-                    vector_basis = basis
-                    degree = basis[0].degree()
-            else:
-                degree = basis[0].degree()
+            degree = len(_all2list(basis[0]))
 
-        # Build an ambient space that fits...
+        # Build an ambient space that fits our matrix basis when
+        # written out as "long vectors."
         V = VectorSpace(field, degree)
 
-        # We overwrite the name "vector_basis" in a second, but never modify it
-        # in place, to this effectively makes a copy of it.
-        deortho_vector_basis = vector_basis
+        # The matrix that will hole the orthonormal -> unorthonormal
+        # coordinate transformation.
         self._deortho_matrix = None
 
         if orthonormalize:
-            from mjo.eja.eja_utils import gram_schmidt
-            if basis_is_matrices:
-                vector_ip = lambda x,y: inner_product(_vec2mat(x), _vec2mat(y))
-                vector_basis = gram_schmidt(vector_basis, vector_ip)
-            else:
-                vector_basis = gram_schmidt(vector_basis, inner_product)
-
-            # Normalize the "matrix" basis, too!
-            basis = vector_basis
+            # Save a copy of the un-orthonormalized basis for later.
+            # Convert it to ambient V (vector) coordinates while we're
+            # at it, because we'd have to do it later anyway.
+            deortho_vector_basis = tuple( V(_all2list(b)) for b in basis )
 
-            if basis_is_matrices:
-                basis = tuple( map(_vec2mat,basis) )
+            from mjo.eja.eja_utils import gram_schmidt
+            basis = tuple(gram_schmidt(basis, inner_product))
 
-        # Save the matrix "basis" for later... this is the last time we'll
-        # reference it in this constructor.
-        if basis_is_matrices:
-            self._matrix_basis = basis
-        else:
-            MS = MatrixSpace(self.base_ring(), degree, 1)
-            self._matrix_basis = tuple( MS(b) for b in basis )
+        # Save the (possibly orthonormalized) matrix basis for
+        # later...
+        self._matrix_basis = basis
 
-        # Now create the vector space for the algebra...
+        # Now create the vector space for the algebra, which will have
+        # its own set of non-ambient coordinates (in terms of the
+        # supplied basis).
+        vector_basis = tuple( V(_all2list(b)) for b in basis )
         W = V.span_of_basis( vector_basis, check=check_axioms)
 
         if orthonormalize:
@@ -182,40 +253,33 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
 
         # Now we actually compute the multiplication and inner-product
         # tables/matrices using the possibly-orthonormalized basis.
-        self._inner_product_matrix = matrix.zero(field, n)
-        self._multiplication_table = [ [0 for j in range(i+1)] for i in range(n) ]
+        self._inner_product_matrix = matrix.identity(field, n)
+        self._multiplication_table = [ [0 for j in range(i+1)]
+                                       for i in range(n) ]
 
-        print("vector_basis:")
-        print(vector_basis)
         # Note: the Jordan and inner-products are defined in terms
         # of the ambient basis. It's important that their arguments
         # are in ambient coordinates as well.
         for i in range(n):
             for j in range(i+1):
                 # ortho basis w.r.t. ambient coords
-                q_i = vector_basis[i]
-                q_j = vector_basis[j]
-
-                if basis_is_matrices:
-                    q_i = _vec2mat(q_i)
-                    q_j = _vec2mat(q_j)
+                q_i = basis[i]
+                q_j = basis[j]
 
+                # The jordan product returns a matrixy answer, so we
+                # have to convert it to the algebra coordinates.
                 elt = jordan_product(q_i, q_j)
-                ip = inner_product(q_i, q_j)
-
-                if basis_is_matrices:
-                    # do another mat2vec because the multiplication
-                    # table is in terms of vectors
-                    elt = _mat2vec(elt)
-
-                # TODO: the jordan product turns things back into
-                # matrices here even if they're supposed to be
-                # vectors. ugh. Can we get rid of vectors all together
-                # please?
-                elt = W.coordinate_vector(elt)
+                elt = W.coordinate_vector(V(_all2list(elt)))
                 self._multiplication_table[i][j] = self.from_vector(elt)
-                self._inner_product_matrix[i,j] = ip
-                self._inner_product_matrix[j,i] = ip
+
+                if not orthonormalize:
+                    # If we're orthonormalizing the basis with respect
+                    # to an inner-product, then the inner-product
+                    # matrix with respect to the resulting basis is
+                    # just going to be the identity.
+                    ip = inner_product(q_i, q_j)
+                    self._inner_product_matrix[i,j] = ip
+                    self._inner_product_matrix[j,i] = ip
 
         self._inner_product_matrix._cache = {'hermitian': True}
         self._inner_product_matrix.set_immutable()
@@ -253,6 +317,35 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
 
 
     def product_on_basis(self, i, j):
+        r"""
+        Returns the Jordan product of the `i` and `j`th basis elements.
+
+        This completely defines the Jordan product on the algebra, and
+        is used direclty by our superclass machinery to implement
+        :meth:`product`.
+
+        SETUP::
+
+            sage: from mjo.eja.eja_algebra import random_eja
+
+        TESTS::
+
+            sage: set_random_seed()
+            sage: J = random_eja()
+            sage: n = J.dimension()
+            sage: bi = J.zero()
+            sage: bj = J.zero()
+            sage: bi_bj = J.zero()*J.zero()
+            sage: if n > 0:
+            ....:     i = ZZ.random_element(n)
+            ....:     j = ZZ.random_element(n)
+            ....:     bi = J.monomial(i)
+            ....:     bj = J.monomial(j)
+            ....:     bi_bj = J.product_on_basis(i,j)
+            sage: bi*bj == bi_bj
+            True
+
+        """
         # We only stored the lower-triangular portion of the
         # multiplication table.
         if j <= i:
@@ -310,11 +403,33 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
             sage: y = J.random_element()
             sage: (n == 1) or (x.inner_product(y) == (x*y).trace()/2)
             True
+
         """
         B = self._inner_product_matrix
         return (B*x.to_vector()).inner_product(y.to_vector())
 
 
+    def is_associative(self):
+        r"""
+        Return whether or not this algebra's Jordan product is associative.
+
+        SETUP::
+
+            sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
+
+        EXAMPLES::
+
+            sage: J = ComplexHermitianEJA(3, field=QQ, orthonormalize=False)
+            sage: J.is_associative()
+            False
+            sage: x = sum(J.gens())
+            sage: A = x.subalgebra_generated_by(orthonormalize=False)
+            sage: A.is_associative()
+            True
+
+        """
+        return "Associative" in self.category().axioms()
+
     def _is_commutative(self):
         r"""
         Whether or not this algebra's multiplication table is commutative.
@@ -323,9 +438,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
         this algebra was constructed with ``check_axioms=False`` and
         passed an invalid multiplication table.
         """
-        return all( self.product_on_basis(i,j) == self.product_on_basis(i,j)
-                    for i in range(self.dimension())
-                    for j in range(self.dimension()) )
+        return all( x*y == y*x for x in self.gens() for y in self.gens() )
 
     def _is_jordanian(self):
         r"""
@@ -344,6 +457,92 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
                     for i in range(self.dimension())
                     for j in range(self.dimension()) )
 
+    def _jordan_product_is_associative(self):
+        r"""
+        Return whether or not this algebra's Jordan product is
+        associative; that is, whether or not `x*(y*z) = (x*y)*z`
+        for all `x,y,x`.
+
+        This method should agree with :meth:`is_associative` unless
+        you lied about the value of the ``associative`` parameter
+        when you constructed the algebra.
+
+        SETUP::
+
+            sage: from mjo.eja.eja_algebra import (random_eja,
+            ....:                                  RealSymmetricEJA,
+            ....:                                  ComplexHermitianEJA,
+            ....:                                  QuaternionHermitianEJA)
+
+        EXAMPLES::
+
+            sage: J = RealSymmetricEJA(4, orthonormalize=False)
+            sage: J._jordan_product_is_associative()
+            False
+            sage: x = sum(J.gens())
+            sage: A = x.subalgebra_generated_by()
+            sage: A._jordan_product_is_associative()
+            True
+
+        ::
+
+            sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
+            sage: J._jordan_product_is_associative()
+            False
+            sage: x = sum(J.gens())
+            sage: A = x.subalgebra_generated_by(orthonormalize=False)
+            sage: A._jordan_product_is_associative()
+            True
+
+        ::
+
+            sage: J = QuaternionHermitianEJA(2)
+            sage: J._jordan_product_is_associative()
+            False
+            sage: x = sum(J.gens())
+            sage: A = x.subalgebra_generated_by()
+            sage: A._jordan_product_is_associative()
+            True
+
+        TESTS:
+
+        The values we've presupplied to the constructors agree with
+        the computation::
+
+            sage: set_random_seed()
+            sage: J = random_eja()
+            sage: J.is_associative() == J._jordan_product_is_associative()
+            True
+
+        """
+        R = self.base_ring()
+
+        # Used to check whether or not something is zero.
+        epsilon = R.zero()
+        if not R.is_exact():
+            # I don't know of any examples that make this magnitude
+            # necessary because I don't know how to make an
+            # associative algebra when the element subalgebra
+            # construction is unreliable (as it is over RDF; we can't
+            # find the degree of an element because we can't compute
+            # the rank of a matrix). But even multiplication of floats
+            # is non-associative, so *some* epsilon is needed... let's
+            # just take the one from _inner_product_is_associative?
+            epsilon = 1e-15
+
+        for i in range(self.dimension()):
+            for j in range(self.dimension()):
+                for k in range(self.dimension()):
+                    x = self.monomial(i)
+                    y = self.monomial(j)
+                    z = self.monomial(k)
+                    diff = (x*y)*z - x*(y*z)
+
+                    if diff.norm() > epsilon:
+                        return False
+
+        return True
+
     def _inner_product_is_associative(self):
         r"""
         Return whether or not this algebra's inner product `B` is
@@ -351,13 +550,16 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
 
         This method should of course always return ``True``, unless
         this algebra was constructed with ``check_axioms=False`` and
-        passed an invalid multiplication table.
+        passed an invalid Jordan or inner-product.
         """
+        R = self.base_ring()
 
-        # Used to check whether or not something is zero in an inexact
-        # ring. This number is sufficient to allow the construction of
-        # QuaternionHermitianEJA(2, field=RDF) with check_axioms=True.
-        epsilon = 1e-16
+        # Used to check whether or not something is zero.
+        epsilon = R.zero()
+        if not R.is_exact():
+            # This choice is sufficient to allow the construction of
+            # QuaternionHermitianEJA(2, field=RDF) with check_axioms=True.
+            epsilon = 1e-15
 
         for i in range(self.dimension()):
             for j in range(self.dimension()):
@@ -367,12 +569,8 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
                     z = self.monomial(k)
                     diff = (x*y).inner_product(z) - x.inner_product(y*z)
 
-                    if self.base_ring().is_exact():
-                        if diff != 0:
-                            return False
-                    else:
-                        if diff.abs() > epsilon:
-                            return False
+                    if diff.abs() > epsilon:
+                        return False
 
         return True
 
@@ -386,7 +584,8 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
 
         SETUP::
 
-            sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+            sage: from mjo.eja.eja_algebra import (random_eja,
+            ....:                                  JordanSpinEJA,
             ....:                                  HadamardEJA,
             ....:                                  RealSymmetricEJA)
 
@@ -408,29 +607,42 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
             ...
             ValueError: not an element of this algebra
 
+        Tuples work as well, provided that the matrix basis for the
+        algebra consists of them::
+
+            sage: J1 = HadamardEJA(3)
+            sage: J2 = RealSymmetricEJA(2)
+            sage: J = cartesian_product([J1,J2])
+            sage: J( (J1.matrix_basis()[1], J2.matrix_basis()[2]) )
+            b1 + b5
+
         TESTS:
 
-        Ensure that we can convert any element of the two non-matrix
-        simple algebras (whose matrix representations are columns)
-        back and forth faithfully::
+        Ensure that we can convert any element back and forth
+        faithfully between its matrix and algebra representations::
 
             sage: set_random_seed()
-            sage: J = HadamardEJA.random_instance()
-            sage: x = J.random_element()
-            sage: J(x.to_vector().column()) == x
-            True
-            sage: J = JordanSpinEJA.random_instance()
+            sage: J = random_eja()
             sage: x = J.random_element()
-            sage: J(x.to_vector().column()) == x
+            sage: J(x.to_matrix()) == x
             True
 
+        We cannot coerce elements between algebras just because their
+        matrix representations are compatible::
+
+            sage: J1 = HadamardEJA(3)
+            sage: J2 = JordanSpinEJA(3)
+            sage: J2(J1.one())
+            Traceback (most recent call last):
+            ...
+            ValueError: not an element of this algebra
+            sage: J1(J2.zero())
+            Traceback (most recent call last):
+            ...
+            ValueError: not an element of this algebra
         """
         msg = "not an element of this algebra"
-        if elt == 0:
-            # The superclass implementation of random_element()
-            # needs to be able to coerce "0" into the algebra.
-            return self.zero()
-        elif elt in self.base_ring():
+        if elt in self.base_ring():
             # Ensure that no base ring -> algebra coercion is performed
             # by this method. There's some stupidity in sage that would
             # otherwise propagate to this method; for example, sage thinks
@@ -438,9 +650,11 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
             raise ValueError(msg)
 
         try:
+            # Try to convert a vector into a column-matrix...
             elt = elt.column()
         except (AttributeError, TypeError):
-            # Try to convert a vector into a column-matrix
+            # and ignore failure, because we weren't really expecting
+            # a vector as an argument anyway.
             pass
 
         if elt not in self.matrix_space():
@@ -453,14 +667,20 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
         # closure whereas the base ring of the 3-by-3 identity matrix
         # could be QQ instead of QQbar.
         #
+        # And, we also have to handle Cartesian product bases (when
+        # the matrix basis consists of tuples) here. The "good news"
+        # is that we're already converting everything to long vectors,
+        # and that strategy works for tuples as well.
+        #
         # We pass check=False because the matrix basis is "guaranteed"
         # to be linearly independent... right? Ha ha.
-        V = VectorSpace(self.base_ring(), elt.nrows()*elt.ncols())
-        W = V.span_of_basis( (_mat2vec(s) for s in self.matrix_basis()),
+        elt = _all2list(elt)
+        V = VectorSpace(self.base_ring(), len(elt))
+        W = V.span_of_basis( (V(_all2list(s)) for s in self.matrix_basis()),
                              check=False)
 
         try:
-            coords =  W.coordinate_vector(_mat2vec(elt))
+            coords = W.coordinate_vector(V(elt))
         except ArithmeticError:  # vector is not in free module
             raise ValueError(msg)
 
@@ -669,15 +889,15 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
             sage: J = JordanSpinEJA(4)
             sage: J.multiplication_table()
             +----++----+----+----+----+
-            | *  || e0 | e1 | e2 | e3 |
+            | *  || b0 | b1 | b2 | b3 |
             +====++====+====+====+====+
-            | e0 || e0 | e1 | e2 | e3 |
+            | b0 || b0 | b1 | b2 | b3 |
             +----++----+----+----+----+
-            | e1 || e1 | e0 | 0  | 0  |
+            | b1 || b1 | b0 | 0  | 0  |
             +----++----+----+----+----+
-            | e2 || e2 | 0  | e0 | 0  |
+            | b2 || b2 | 0  | b0 | 0  |
             +----++----+----+----+----+
-            | e3 || e3 | 0  | 0  | e0 |
+            | b3 || b3 | 0  | 0  | b0 |
             +----++----+----+----+----+
 
         """
@@ -687,8 +907,8 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
 
         # And to each subsequent row, prepend an entry that belongs to
         # the left-side "header column."
-        M += [ [self.monomial(i)] + [ self.product_on_basis(i,j)
-                                      for j in range(n) ]
+        M += [ [self.monomial(i)] + [ self.monomial(i)*self.monomial(j)
+                                    for j in range(n) ]
                for i in range(n) ]
 
         return table(M, header_row=True, header_column=True, frame=True)
@@ -718,7 +938,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
         Why implement this for non-matrix algebras? Avoiding special
         cases for the :class:`BilinearFormEJA` pays with simplicity in
         its own right. But mainly, we would like to be able to assume
-        that elements of a :class:`DirectSumEJA` can be displayed
+        that elements of a :class:`CartesianProductEJA` can be displayed
         nicely, without having to have special classes for direct sums
         one of whose components was a matrix algebra.
 
@@ -731,7 +951,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
 
             sage: J = RealSymmetricEJA(2)
             sage: J.basis()
-            Finite family {0: e0, 1: e1, 2: e2}
+            Finite family {0: b0, 1: b1, 2: b2}
             sage: J.matrix_basis()
             (
             [1 0]  [                  0 0.7071067811865475?]  [0 0]
@@ -742,7 +962,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
 
             sage: J = JordanSpinEJA(2)
             sage: J.basis()
-            Finite family {0: e0, 1: e1}
+            Finite family {0: b0, 1: b1}
             sage: J.matrix_basis()
             (
             [1]  [0]
@@ -758,17 +978,54 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
         we think of them as matrices (including column vectors of the
         appropriate size).
 
-        Generally this will be an `n`-by-`1` column-vector space,
+        "By default" this will be an `n`-by-`1` column-matrix space,
         except when the algebra is trivial. There it's `n`-by-`n`
         (where `n` is zero), to ensure that two elements of the matrix
-        space (empty matrices) can be multiplied.
+        space (empty matrices) can be multiplied. For algebras of
+        matrices, this returns the space in which their
+        real embeddings live.
+
+        SETUP::
+
+            sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+            ....:                                  JordanSpinEJA,
+            ....:                                  QuaternionHermitianEJA,
+            ....:                                  TrivialEJA)
+
+        EXAMPLES:
+
+        By default, the matrix representation is just a column-matrix
+        equivalent to the vector representation::
+
+            sage: J = JordanSpinEJA(3)
+            sage: J.matrix_space()
+            Full MatrixSpace of 3 by 1 dense matrices over Algebraic
+            Real Field
+
+        The matrix representation in the trivial algebra is
+        zero-by-zero instead of the usual `n`-by-one::
+
+            sage: J = TrivialEJA()
+            sage: J.matrix_space()
+            Full MatrixSpace of 0 by 0 dense matrices over Algebraic
+            Real Field
+
+        The matrix space for complex/quaternion Hermitian matrix EJA
+        is the space in which their real-embeddings live, not the
+        original complex/quaternion matrix space::
+
+            sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
+            sage: J.matrix_space()
+            Full MatrixSpace of 4 by 4 dense matrices over Rational Field
+            sage: J = QuaternionHermitianEJA(1,field=QQ,orthonormalize=False)
+            sage: J.matrix_space()
+            Full MatrixSpace of 4 by 4 dense matrices over Rational Field
 
-        Matrix algebras override this with something more useful.
         """
         if self.is_trivial():
             return MatrixSpace(self.base_ring(), 0)
         else:
-            return self._matrix_basis[0].matrix_space()
+            return self.matrix_basis()[0].parent()
 
 
     @cached_method
@@ -781,23 +1038,57 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
             sage: from mjo.eja.eja_algebra import (HadamardEJA,
             ....:                                  random_eja)
 
-        EXAMPLES::
+        EXAMPLES:
+
+        We can compute unit element in the Hadamard EJA::
+
+            sage: J = HadamardEJA(5)
+            sage: J.one()
+            b0 + b1 + b2 + b3 + b4
+
+        The unit element in the Hadamard EJA is inherited in the
+        subalgebras generated by its elements::
 
             sage: J = HadamardEJA(5)
             sage: J.one()
-            e0 + e1 + e2 + e3 + e4
+            b0 + b1 + b2 + b3 + b4
+            sage: x = sum(J.gens())
+            sage: A = x.subalgebra_generated_by(orthonormalize=False)
+            sage: A.one()
+            c0
+            sage: A.one().superalgebra_element()
+            b0 + b1 + b2 + b3 + b4
 
         TESTS:
 
-        The identity element acts like the identity::
+        The identity element acts like the identity, regardless of
+        whether or not we orthonormalize::
 
             sage: set_random_seed()
             sage: J = random_eja()
             sage: x = J.random_element()
             sage: J.one()*x == x and x*J.one() == x
             True
+            sage: A = x.subalgebra_generated_by()
+            sage: y = A.random_element()
+            sage: A.one()*y == y and y*A.one() == y
+            True
+
+        ::
+
+            sage: set_random_seed()
+            sage: J = random_eja(field=QQ, orthonormalize=False)
+            sage: x = J.random_element()
+            sage: J.one()*x == x and x*J.one() == x
+            True
+            sage: A = x.subalgebra_generated_by(orthonormalize=False)
+            sage: y = A.random_element()
+            sage: A.one()*y == y and y*A.one() == y
+            True
 
-        The matrix of the unit element's operator is the identity::
+        The matrix of the unit element's operator is the identity,
+        regardless of the base field and whether or not we
+        orthonormalize::
 
             sage: set_random_seed()
             sage: J = random_eja()
@@ -805,6 +1096,27 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
             sage: expected = matrix.identity(J.base_ring(), J.dimension())
             sage: actual == expected
             True
+            sage: x = J.random_element()
+            sage: A = x.subalgebra_generated_by()
+            sage: actual = A.one().operator().matrix()
+            sage: expected = matrix.identity(A.base_ring(), A.dimension())
+            sage: actual == expected
+            True
+
+        ::
+
+            sage: set_random_seed()
+            sage: J = random_eja(field=QQ, orthonormalize=False)
+            sage: actual = J.one().operator().matrix()
+            sage: expected = matrix.identity(J.base_ring(), J.dimension())
+            sage: actual == expected
+            True
+            sage: x = J.random_element()
+            sage: A = x.subalgebra_generated_by(orthonormalize=False)
+            sage: actual = A.one().operator().matrix()
+            sage: expected = matrix.identity(A.base_ring(), A.dimension())
+            sage: actual == expected
+            True
 
         Ensure that the cached unit element (often precomputed by
         hand) agrees with the computed one::
@@ -816,6 +1128,15 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
             sage: J.one() == cached
             True
 
+        ::
+
+            sage: set_random_seed()
+            sage: J = random_eja(field=QQ, orthonormalize=False)
+            sage: cached = J.one()
+            sage: J.one.clear_cache()
+            sage: J.one() == cached
+            True
+
         """
         # We can brute-force compute the matrices of the operators
         # that correspond to the basis elements of this algebra.
@@ -960,14 +1281,12 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
         if not c.is_idempotent():
             raise ValueError("element is not idempotent: %s" % c)
 
-        from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra
-
         # Default these to what they should be if they turn out to be
         # trivial, because eigenspaces_left() won't return eigenvalues
         # corresponding to trivial spaces (e.g. it returns only the
         # eigenspace corresponding to lambda=1 if you take the
         # decomposition relative to the identity element).
-        trivial = FiniteDimensionalEJASubalgebra(self, ())
+        trivial = self.subalgebra(())
         J0 = trivial                          # eigenvalue zero
         J5 = VectorSpace(self.base_ring(), 0) # eigenvalue one-half
         J1 = trivial                          # eigenvalue one
@@ -977,9 +1296,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
                 J5 = eigspace
             else:
                 gens = tuple( self.from_vector(b) for b in eigspace.basis() )
-                subalg = FiniteDimensionalEJASubalgebra(self,
-                                                        gens,
-                                                        check_axioms=False)
+                subalg = self.subalgebra(gens, check_axioms=False)
                 if eigval == 0:
                     J0 = subalg
                 elif eigval == 1:
@@ -1068,6 +1385,21 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
         r"""
         The `r` polynomial coefficients of the "characteristic polynomial
         of" function.
+
+        SETUP::
+
+            sage: from mjo.eja.eja_algebra import random_eja
+
+        TESTS:
+
+        The theory shows that these are all homogeneous polynomials of
+        a known degree::
+
+            sage: set_random_seed()
+            sage: J = random_eja()
+            sage: all(p.is_homogeneous() for p in J._charpoly_coefficients())
+            True
+
         """
         n = self.dimension()
         R = self.coordinate_polynomial_ring()
@@ -1103,10 +1435,17 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
 
         # The theory says that only the first "r" coefficients are
         # nonzero, and they actually live in the original polynomial
-        # ring and not the fraction field. We negate them because
-        # in the actual characteristic polynomial, they get moved
-        # to the other side where x^r lives.
-        return -A_rref.solve_right(E*b).change_ring(R)[:r]
+        # ring and not the fraction field. We negate them because in
+        # the actual characteristic polynomial, they get moved to the
+        # other side where x^r lives. We don't bother to trim A_rref
+        # down to a square matrix and solve the resulting system,
+        # because the upper-left r-by-r portion of A_rref is
+        # guaranteed to be the identity matrix, so e.g.
+        #
+        #   A_rref.solve_right(Y)
+        #
+        # would just be returning Y.
+        return (-E*b)[:r].change_ring(R)
 
     @cached_method
     def rank(self):
@@ -1167,7 +1506,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
 
             sage: set_random_seed()    # long time
             sage: J = random_eja()     # long time
-            sage: caches = J.rank()    # long time
+            sage: cached = J.rank()    # long time
             sage: J.rank.clear_cache() # long time
             sage: J.rank() == cached   # long time
             True
@@ -1176,6 +1515,14 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
         return len(self._charpoly_coefficients())
 
 
+    def subalgebra(self, basis, **kwargs):
+        r"""
+        Create a subalgebra of this algebra from the given basis.
+        """
+        from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra
+        return FiniteDimensionalEJASubalgebra(self, basis, **kwargs)
+
+
     def vector_space(self):
         """
         Return the vector space that underlies this algebra.
@@ -1194,11 +1541,10 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
         return self.zero().to_vector().parent().ambient_vector_space()
 
 
-    Element = FiniteDimensionalEJAElement
 
 class RationalBasisEJA(FiniteDimensionalEJA):
     r"""
-    New class for algebras whose supplied basis elements have all rational entries.
+    Algebras whose supplied basis elements have all rational entries.
 
     SETUP::
 
@@ -1223,9 +1569,7 @@ class RationalBasisEJA(FiniteDimensionalEJA):
                  jordan_product,
                  inner_product,
                  field=AA,
-                 orthonormalize=True,
                  check_field=True,
-                 check_axioms=True,
                  **kwargs):
 
         if check_field:
@@ -1234,6 +1578,14 @@ class RationalBasisEJA(FiniteDimensionalEJA):
             if not all( all(b_i in QQ for b_i in b.list()) for b in basis ):
                 raise TypeError("basis not rational")
 
+        super().__init__(basis,
+                         jordan_product,
+                         inner_product,
+                         field=field,
+                         check_field=check_field,
+                         **kwargs)
+
+        self._rational_algebra = None
         if field is not QQ:
             # There's no point in constructing the extra algebra if this
             # one is already rational.
@@ -1246,18 +1598,10 @@ class RationalBasisEJA(FiniteDimensionalEJA):
                                        jordan_product,
                                        inner_product,
                                        field=QQ,
+                                       associative=self.is_associative(),
                                        orthonormalize=False,
                                        check_field=False,
-                                       check_axioms=False,
-                                       **kwargs)
-
-        super().__init__(basis,
-                         jordan_product,
-                         inner_product,
-                         field=field,
-                         check_field=check_field,
-                         check_axioms=check_axioms,
-                         **kwargs)
+                                       check_axioms=False)
 
     @cached_method
     def _charpoly_coefficients(self):
@@ -1296,7 +1640,14 @@ class RationalBasisEJA(FiniteDimensionalEJA):
         a = ( a_i.change_ring(self.base_ring())
               for a_i in self._rational_algebra._charpoly_coefficients() )
 
-        # Now convert the coordinate variables back to the
+        if self._deortho_matrix is None:
+            # This can happen if our base ring was, say, AA and we
+            # chose not to (or didn't need to) orthonormalize. It's
+            # still faster to do the computations over QQ even if
+            # the numbers in the boxes stay the same.
+            return tuple(a)
+
+        # Otherwise, convert the coordinate variables back to the
         # deorthonormalized ones.
         R = self.coordinate_polynomial_ring()
         from sage.modules.free_module_element import vector
@@ -1306,7 +1657,7 @@ class RationalBasisEJA(FiniteDimensionalEJA):
         subs_dict = { X[i]: BX[i] for i in range(len(X)) }
         return tuple( a_i.subs(subs_dict) for a_i in a )
 
-class ConcreteEJA(RationalBasisEJA):
+class ConcreteEJA(FiniteDimensionalEJA):
     r"""
     A class for the Euclidean Jordan algebras that we know by name.
 
@@ -1375,6 +1726,19 @@ class ConcreteEJA(RationalBasisEJA):
 
 
 class MatrixEJA:
+    @staticmethod
+    def jordan_product(X,Y):
+        return (X*Y + Y*X)/2
+
+    @staticmethod
+    def trace_inner_product(X,Y):
+        r"""
+        A trace inner-product for matrices that aren't embedded in the
+        reals. It takes MATRICES as arguments, not EJA elements.
+        """
+        return (X*Y).trace().real()
+
+class RealEmbeddedMatrixEJA(MatrixEJA):
     @staticmethod
     def dimension_over_reals():
         r"""
@@ -1420,9 +1784,6 @@ class MatrixEJA:
             raise ValueError("the matrix 'M' must be a real embedding")
         return M
 
-    @staticmethod
-    def jordan_product(X,Y):
-        return (X*Y + Y*X)/2
 
     @classmethod
     def trace_inner_product(cls,X,Y):
@@ -1431,29 +1792,11 @@ class MatrixEJA:
 
         SETUP::
 
-            sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
-            ....:                                  ComplexHermitianEJA,
+            sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
             ....:                                  QuaternionHermitianEJA)
 
         EXAMPLES::
 
-        This gives the same answer as it would if we computed the trace
-        from the unembedded (original) matrices::
-
-            sage: set_random_seed()
-            sage: J = RealSymmetricEJA.random_instance()
-            sage: x,y = J.random_elements(2)
-            sage: Xe = x.to_matrix()
-            sage: Ye = y.to_matrix()
-            sage: X = J.real_unembed(Xe)
-            sage: Y = J.real_unembed(Ye)
-            sage: expected = (X*Y).trace()
-            sage: actual = J.trace_inner_product(Xe,Ye)
-            sage: actual == expected
-            True
-
-        ::
-
             sage: set_random_seed()
             sage: J = ComplexHermitianEJA.random_instance()
             sage: x,y = J.random_elements(2)
@@ -1481,27 +1824,15 @@ class MatrixEJA:
             True
 
         """
-        Xu = cls.real_unembed(X)
-        Yu = cls.real_unembed(Y)
-        tr = (Xu*Yu).trace()
-
-        try:
-            # Works in QQ, AA, RDF, et cetera.
-            return tr.real()
-        except AttributeError:
-            # A quaternion doesn't have a real() method, but does
-            # have coefficient_tuple() method that returns the
-            # coefficients of 1, i, j, and k -- in that order.
-            return tr.coefficient_tuple()[0]
-
-
-class RealMatrixEJA(MatrixEJA):
-    @staticmethod
-    def dimension_over_reals():
-        return 1
+        # This does in fact compute the real part of the trace.
+        # If we compute the trace of e.g. a complex matrix M,
+        # then we do so by adding up its diagonal entries --
+        # call them z_1 through z_n. The real embedding of z_1
+        # will be a 2-by-2 REAL matrix [a, b; -b, a] whose trace
+        # as a REAL matrix will be 2*a = 2*Re(z_1). And so forth.
+        return (X*Y).trace()/cls.dimension_over_reals()
 
-
-class RealSymmetricEJA(ConcreteEJA, RealMatrixEJA):
+class RealSymmetricEJA(ConcreteEJA, RationalBasisEJA, MatrixEJA):
     """
     The rank-n simple EJA consisting of real symmetric n-by-n
     matrices, the usual symmetric Jordan product, and the trace inner
@@ -1514,19 +1845,19 @@ class RealSymmetricEJA(ConcreteEJA, RealMatrixEJA):
     EXAMPLES::
 
         sage: J = RealSymmetricEJA(2)
-        sage: e0, e1, e2 = J.gens()
-        sage: e0*e0
-        e0
-        sage: e1*e1
-        1/2*e0 + 1/2*e2
-        sage: e2*e2
-        e2
+        sage: b0, b1, b2 = J.gens()
+        sage: b0*b0
+        b0
+        sage: b1*b1
+        1/2*b0 + 1/2*b2
+        sage: b2*b2
+        b2
 
     In theory, our "field" can be any subfield of the reals::
 
-        sage: RealSymmetricEJA(2, field=RDF)
+        sage: RealSymmetricEJA(2, field=RDF, check_axioms=True)
         Euclidean Jordan algebra of dimension 3 over Real Double Field
-        sage: RealSymmetricEJA(2, field=RR)
+        sage: RealSymmetricEJA(2, field=RR, check_axioms=True)
         Euclidean Jordan algebra of dimension 3 over Real Field with
         53 bits of precision
 
@@ -1567,7 +1898,7 @@ class RealSymmetricEJA(ConcreteEJA, RealMatrixEJA):
 
     """
     @classmethod
-    def _denormalized_basis(cls, n):
+    def _denormalized_basis(cls, n, field):
         """
         Return a basis for the space of real symmetric n-by-n matrices.
 
@@ -1579,7 +1910,7 @@ class RealSymmetricEJA(ConcreteEJA, RealMatrixEJA):
 
             sage: set_random_seed()
             sage: n = ZZ.random_element(1,5)
-            sage: B = RealSymmetricEJA._denormalized_basis(n)
+            sage: B = RealSymmetricEJA._denormalized_basis(n,ZZ)
             sage: all( M.is_symmetric() for M in  B)
             True
 
@@ -1589,7 +1920,7 @@ class RealSymmetricEJA(ConcreteEJA, RealMatrixEJA):
         S = []
         for i in range(n):
             for j in range(i+1):
-                Eij = matrix(ZZ, n, lambda k,l: k==i and l==j)
+                Eij = matrix(field, n, lambda k,l: k==i and l==j)
                 if i == j:
                     Sij = Eij
                 else:
@@ -1610,29 +1941,67 @@ class RealSymmetricEJA(ConcreteEJA, RealMatrixEJA):
         n = ZZ.random_element(cls._max_random_instance_size() + 1)
         return cls(n, **kwargs)
 
-    def __init__(self, n, **kwargs):
+    def __init__(self, n, field=AA, **kwargs):
         # We know this is a valid EJA, but will double-check
         # if the user passes check_axioms=True.
         if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
 
-        super(RealSymmetricEJA, self).__init__(self._denormalized_basis(n),
-                                               self.jordan_product,
-                                               self.trace_inner_product,
-                                               **kwargs)
+        associative = False
+        if n <= 1:
+            associative = True
+
+        super().__init__(self._denormalized_basis(n,field),
+                         self.jordan_product,
+                         self.trace_inner_product,
+                         field=field,
+                         associative=associative,
+                         **kwargs)
 
         # TODO: this could be factored out somehow, but is left here
         # because the MatrixEJA is not presently a subclass of the
         # FDEJA class that defines rank() and one().
         self.rank.set_cache(n)
-        idV = matrix.identity(ZZ, self.dimension_over_reals()*n)
+        idV = self.matrix_space().one()
         self.one.set_cache(self(idV))
 
 
 
-class ComplexMatrixEJA(MatrixEJA):
-    @staticmethod
-    def dimension_over_reals():
-        return 2
+class ComplexMatrixEJA(RealEmbeddedMatrixEJA):
+    # A manual dictionary-cache for the complex_extension() method,
+    # since apparently @classmethods can't also be @cached_methods.
+    _complex_extension = {}
+
+    @classmethod
+    def complex_extension(cls,field):
+        r"""
+        The complex field that we embed/unembed, as an extension
+        of the given ``field``.
+        """
+        if field in cls._complex_extension:
+            return cls._complex_extension[field]
+
+        # Sage doesn't know how to adjoin the complex "i" (the root of
+        # x^2 + 1) to a field in a general way. Here, we just enumerate
+        # all of the cases that I have cared to support so far.
+        if field is AA:
+            # Sage doesn't know how to embed AA into QQbar, i.e. how
+            # to adjoin sqrt(-1) to AA.
+            F = QQbar
+        elif not field.is_exact():
+            # RDF or RR
+            F = field.complex_field()
+        else:
+            # Works for QQ and... maybe some other fields.
+            R = PolynomialRing(field, 'z')
+            z = R.gen()
+            F = field.extension(z**2 + 1, 'I', embedding=CLF(-1).sqrt())
+
+        cls._complex_extension[field] = F
+        return F
+
+    @staticmethod
+    def dimension_over_reals():
+        return 2
 
     @classmethod
     def real_embed(cls,M):
@@ -1676,7 +2045,7 @@ class ComplexMatrixEJA(MatrixEJA):
             True
 
         """
-        super(ComplexMatrixEJA,cls).real_embed(M)
+        super().real_embed(M)
         n = M.nrows()
 
         # We don't need any adjoined elements...
@@ -1684,9 +2053,10 @@ class ComplexMatrixEJA(MatrixEJA):
 
         blocks = []
         for z in M.list():
-            a = z.list()[0] # real part, I guess
-            b = z.list()[1] # imag part, I guess
-            blocks.append(matrix(field, 2, [[a,b],[-b,a]]))
+            a = z.real()
+            b = z.imag()
+            blocks.append(matrix(field, 2, [ [ a, b],
+                                             [-b, a] ]))
 
         return matrix.block(field, n, blocks)
 
@@ -1722,29 +2092,10 @@ class ComplexMatrixEJA(MatrixEJA):
             True
 
         """
-        super(ComplexMatrixEJA,cls).real_unembed(M)
+        super().real_unembed(M)
         n = ZZ(M.nrows())
         d = cls.dimension_over_reals()
-
-        # If "M" was normalized, its base ring might have roots
-        # adjoined and they can stick around after unembedding.
-        field = M.base_ring()
-        R = PolynomialRing(field, 'z')
-        z = R.gen()
-
-        # Sage doesn't know how to adjoin the complex "i" (the root of
-        # x^2 + 1) to a field in a general way. Here, we just enumerate
-        # all of the cases that I have cared to support so far.
-        if field is AA:
-            # Sage doesn't know how to embed AA into QQbar, i.e. how
-            # to adjoin sqrt(-1) to AA.
-            F = QQbar
-        elif not field.is_exact():
-            # RDF or RR
-            F = field.complex_field()
-        else:
-            # Works for QQ and... maybe some other fields.
-            F = field.extension(z**2 + 1, 'I', embedding=CLF(-1).sqrt())
+        F = cls.complex_extension(M.base_ring())
         i = F.gen()
 
         # Go top-left to bottom-right (reading order), converting every
@@ -1763,7 +2114,7 @@ class ComplexMatrixEJA(MatrixEJA):
         return matrix(F, n/d, elements)
 
 
-class ComplexHermitianEJA(ConcreteEJA, ComplexMatrixEJA):
+class ComplexHermitianEJA(ConcreteEJA, RationalBasisEJA, ComplexMatrixEJA):
     """
     The rank-n simple EJA consisting of complex Hermitian n-by-n
     matrices over the real numbers, the usual symmetric Jordan product,
@@ -1778,9 +2129,9 @@ class ComplexHermitianEJA(ConcreteEJA, ComplexMatrixEJA):
 
     In theory, our "field" can be any subfield of the reals::
 
-        sage: ComplexHermitianEJA(2, field=RDF)
+        sage: ComplexHermitianEJA(2, field=RDF, check_axioms=True)
         Euclidean Jordan algebra of dimension 4 over Real Double Field
-        sage: ComplexHermitianEJA(2, field=RR)
+        sage: ComplexHermitianEJA(2, field=RR, check_axioms=True)
         Euclidean Jordan algebra of dimension 4 over Real Field with
         53 bits of precision
 
@@ -1822,7 +2173,7 @@ class ComplexHermitianEJA(ConcreteEJA, ComplexMatrixEJA):
     """
 
     @classmethod
-    def _denormalized_basis(cls, n):
+    def _denormalized_basis(cls, n, field):
         """
         Returns a basis for the space of complex Hermitian n-by-n matrices.
 
@@ -1840,16 +2191,14 @@ class ComplexHermitianEJA(ConcreteEJA, ComplexMatrixEJA):
 
             sage: set_random_seed()
             sage: n = ZZ.random_element(1,5)
-            sage: field = QuadraticField(2, 'sqrt2')
-            sage: B = ComplexHermitianEJA._denormalized_basis(n)
+            sage: B = ComplexHermitianEJA._denormalized_basis(n,ZZ)
             sage: all( M.is_symmetric() for M in  B)
             True
 
         """
-        field = ZZ
-        R = PolynomialRing(field, 'z')
+        R = PolynomialRing(ZZ, 'z')
         z = R.gen()
-        F = field.extension(z**2 + 1, 'I')
+        F = ZZ.extension(z**2 + 1, 'I')
         I = F.gen(1)
 
         # This is like the symmetric case, but we need to be careful:
@@ -1858,33 +2207,48 @@ class ComplexHermitianEJA(ConcreteEJA, ComplexMatrixEJA):
         #   * The diagonal will (as a result) be real.
         #
         S = []
+        Eij = matrix.zero(F,n)
         for i in range(n):
             for j in range(i+1):
-                Eij = matrix(F, n, lambda k,l: k==i and l==j)
+                # "build" E_ij
+                Eij[i,j] = 1
                 if i == j:
                     Sij = cls.real_embed(Eij)
                     S.append(Sij)
                 else:
                     # The second one has a minus because it's conjugated.
-                    Sij_real = cls.real_embed(Eij + Eij.transpose())
+                    Eij[j,i] = 1 # Eij = Eij + Eij.transpose()
+                    Sij_real = cls.real_embed(Eij)
                     S.append(Sij_real)
-                    Sij_imag = cls.real_embed(I*Eij - I*Eij.transpose())
+                    # Eij = I*Eij - I*Eij.transpose()
+                    Eij[i,j] = I
+                    Eij[j,i] = -I
+                    Sij_imag = cls.real_embed(Eij)
                     S.append(Sij_imag)
+                    Eij[j,i] = 0
+                # "erase" E_ij
+                Eij[i,j] = 0
 
-        # Since we embedded these, we can drop back to the "field" that we
-        # started with instead of the complex extension "F".
+        # Since we embedded the entries, we can drop back to the
+        # desired real "field" instead of the extension "F".
         return tuple( s.change_ring(field) for s in S )
 
 
-    def __init__(self, n, **kwargs):
+    def __init__(self, n, field=AA, **kwargs):
         # We know this is a valid EJA, but will double-check
         # if the user passes check_axioms=True.
         if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
 
-        super(ComplexHermitianEJA, self).__init__(self._denormalized_basis(n),
-                                                  self.jordan_product,
-                                                  self.trace_inner_product,
-                                                  **kwargs)
+        associative = False
+        if n <= 1:
+            associative = True
+
+        super().__init__(self._denormalized_basis(n,field),
+                         self.jordan_product,
+                         self.trace_inner_product,
+                         field=field,
+                         associative=associative,
+                         **kwargs)
         # TODO: this could be factored out somehow, but is left here
         # because the MatrixEJA is not presently a subclass of the
         # FDEJA class that defines rank() and one().
@@ -1904,7 +2268,26 @@ class ComplexHermitianEJA(ConcreteEJA, ComplexMatrixEJA):
         n = ZZ.random_element(cls._max_random_instance_size() + 1)
         return cls(n, **kwargs)
 
-class QuaternionMatrixEJA(MatrixEJA):
+class QuaternionMatrixEJA(RealEmbeddedMatrixEJA):
+
+    # A manual dictionary-cache for the quaternion_extension() method,
+    # since apparently @classmethods can't also be @cached_methods.
+    _quaternion_extension = {}
+
+    @classmethod
+    def quaternion_extension(cls,field):
+        r"""
+        The quaternion field that we embed/unembed, as an extension
+        of the given ``field``.
+        """
+        if field in cls._quaternion_extension:
+            return cls._quaternion_extension[field]
+
+        Q = QuaternionAlgebra(field,-1,-1)
+
+        cls._quaternion_extension[field] = Q
+        return Q
+
     @staticmethod
     def dimension_over_reals():
         return 4
@@ -1948,7 +2331,7 @@ class QuaternionMatrixEJA(MatrixEJA):
             True
 
         """
-        super(QuaternionMatrixEJA,cls).real_embed(M)
+        super().real_embed(M)
         quaternions = M.base_ring()
         n = M.nrows()
 
@@ -2003,14 +2386,13 @@ class QuaternionMatrixEJA(MatrixEJA):
             True
 
         """
-        super(QuaternionMatrixEJA,cls).real_unembed(M)
+        super().real_unembed(M)
         n = ZZ(M.nrows())
         d = cls.dimension_over_reals()
 
         # Use the base ring of the matrix to ensure that its entries can be
         # multiplied by elements of the quaternion algebra.
-        field = M.base_ring()
-        Q = QuaternionAlgebra(field,-1,-1)
+        Q = cls.quaternion_extension(M.base_ring())
         i,j,k = Q.gens()
 
         # Go top-left to bottom-right (reading order), converting every
@@ -2034,7 +2416,9 @@ class QuaternionMatrixEJA(MatrixEJA):
         return matrix(Q, n/d, elements)
 
 
-class QuaternionHermitianEJA(ConcreteEJA, QuaternionMatrixEJA):
+class QuaternionHermitianEJA(ConcreteEJA,
+                             RationalBasisEJA,
+                             QuaternionMatrixEJA):
     r"""
     The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
     matrices, the usual symmetric Jordan product, and the
@@ -2049,9 +2433,9 @@ class QuaternionHermitianEJA(ConcreteEJA, QuaternionMatrixEJA):
 
     In theory, our "field" can be any subfield of the reals::
 
-        sage: QuaternionHermitianEJA(2, field=RDF)
+        sage: QuaternionHermitianEJA(2, field=RDF, check_axioms=True)
         Euclidean Jordan algebra of dimension 6 over Real Double Field
-        sage: QuaternionHermitianEJA(2, field=RR)
+        sage: QuaternionHermitianEJA(2, field=RR, check_axioms=True)
         Euclidean Jordan algebra of dimension 6 over Real Field with
         53 bits of precision
 
@@ -2092,7 +2476,7 @@ class QuaternionHermitianEJA(ConcreteEJA, QuaternionMatrixEJA):
 
     """
     @classmethod
-    def _denormalized_basis(cls, n):
+    def _denormalized_basis(cls, n, field):
         """
         Returns a basis for the space of quaternion Hermitian n-by-n matrices.
 
@@ -2110,12 +2494,11 @@ class QuaternionHermitianEJA(ConcreteEJA, QuaternionMatrixEJA):
 
             sage: set_random_seed()
             sage: n = ZZ.random_element(1,5)
-            sage: B = QuaternionHermitianEJA._denormalized_basis(n)
+            sage: B = QuaternionHermitianEJA._denormalized_basis(n,ZZ)
             sage: all( M.is_symmetric() for M in B )
             True
 
         """
-        field = ZZ
         Q = QuaternionAlgebra(QQ,-1,-1)
         I,J,K = Q.gens()
 
@@ -2125,38 +2508,61 @@ class QuaternionHermitianEJA(ConcreteEJA, QuaternionMatrixEJA):
         #   * The diagonal will (as a result) be real.
         #
         S = []
+        Eij = matrix.zero(Q,n)
         for i in range(n):
             for j in range(i+1):
-                Eij = matrix(Q, n, lambda k,l: k==i and l==j)
+                # "build" E_ij
+                Eij[i,j] = 1
                 if i == j:
                     Sij = cls.real_embed(Eij)
                     S.append(Sij)
                 else:
                     # The second, third, and fourth ones have a minus
                     # because they're conjugated.
-                    Sij_real = cls.real_embed(Eij + Eij.transpose())
+                    # Eij = Eij + Eij.transpose()
+                    Eij[j,i] = 1
+                    Sij_real = cls.real_embed(Eij)
                     S.append(Sij_real)
-                    Sij_I = cls.real_embed(I*Eij - I*Eij.transpose())
+                    # Eij = I*(Eij - Eij.transpose())
+                    Eij[i,j] = I
+                    Eij[j,i] = -I
+                    Sij_I = cls.real_embed(Eij)
                     S.append(Sij_I)
-                    Sij_J = cls.real_embed(J*Eij - J*Eij.transpose())
+                    # Eij = J*(Eij - Eij.transpose())
+                    Eij[i,j] = J
+                    Eij[j,i] = -J
+                    Sij_J = cls.real_embed(Eij)
                     S.append(Sij_J)
-                    Sij_K = cls.real_embed(K*Eij - K*Eij.transpose())
+                    # Eij = K*(Eij - Eij.transpose())
+                    Eij[i,j] = K
+                    Eij[j,i] = -K
+                    Sij_K = cls.real_embed(Eij)
                     S.append(Sij_K)
+                    Eij[j,i] = 0
+                # "erase" E_ij
+                Eij[i,j] = 0
 
-        # Since we embedded these, we can drop back to the "field" that we
-        # started with instead of the quaternion algebra "Q".
+        # Since we embedded the entries, we can drop back to the
+        # desired real "field" instead of the quaternion algebra "Q".
         return tuple( s.change_ring(field) for s in S )
 
 
-    def __init__(self, n, **kwargs):
+    def __init__(self, n, field=AA, **kwargs):
         # We know this is a valid EJA, but will double-check
         # if the user passes check_axioms=True.
         if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
 
-        super(QuaternionHermitianEJA, self).__init__(self._denormalized_basis(n),
-                                                     self.jordan_product,
-                                                     self.trace_inner_product,
-                                                     **kwargs)
+        associative = False
+        if n <= 1:
+            associative = True
+
+        super().__init__(self._denormalized_basis(n,field),
+                         self.jordan_product,
+                         self.trace_inner_product,
+                         field=field,
+                         associative=associative,
+                         **kwargs)
+
         # TODO: this could be factored out somehow, but is left here
         # because the MatrixEJA is not presently a subclass of the
         # FDEJA class that defines rank() and one().
@@ -2180,8 +2586,215 @@ class QuaternionHermitianEJA(ConcreteEJA, QuaternionMatrixEJA):
         n = ZZ.random_element(cls._max_random_instance_size() + 1)
         return cls(n, **kwargs)
 
+class OctonionHermitianEJA(ConcreteEJA, MatrixEJA):
+    r"""
+    SETUP::
+
+        sage: from mjo.eja.eja_algebra import (FiniteDimensionalEJA,
+        ....:                                  OctonionHermitianEJA)
+
+    EXAMPLES:
+
+    The 3-by-3 algebra satisfies the axioms of an EJA::
+
+        sage: OctonionHermitianEJA(3,                    # long time
+        ....:                      field=QQ,             # long time
+        ....:                      orthonormalize=False, # long time
+        ....:                      check_axioms=True)    # long time
+        Euclidean Jordan algebra of dimension 27 over Rational Field
+
+    After a change-of-basis, the 2-by-2 algebra has the same
+    multiplication table as the ten-dimensional Jordan spin algebra::
+
+        sage: b = OctonionHermitianEJA._denormalized_basis(2,QQ)
+        sage: basis = (b[0] + b[9],) + b[1:9] + (b[0] - b[9],)
+        sage: jp = OctonionHermitianEJA.jordan_product
+        sage: ip = OctonionHermitianEJA.trace_inner_product
+        sage: J = FiniteDimensionalEJA(basis,
+        ....:                          jp,
+        ....:                          ip,
+        ....:                          field=QQ,
+        ....:                          orthonormalize=False)
+        sage: J.multiplication_table()
+        +----++----+----+----+----+----+----+----+----+----+----+
+        | *  || b0 | b1 | b2 | b3 | b4 | b5 | b6 | b7 | b8 | b9 |
+        +====++====+====+====+====+====+====+====+====+====+====+
+        | b0 || b0 | b1 | b2 | b3 | b4 | b5 | b6 | b7 | b8 | b9 |
+        +----++----+----+----+----+----+----+----+----+----+----+
+        | b1 || b1 | b0 | 0  | 0  | 0  | 0  | 0  | 0  | 0  | 0  |
+        +----++----+----+----+----+----+----+----+----+----+----+
+        | b2 || b2 | 0  | b0 | 0  | 0  | 0  | 0  | 0  | 0  | 0  |
+        +----++----+----+----+----+----+----+----+----+----+----+
+        | b3 || b3 | 0  | 0  | b0 | 0  | 0  | 0  | 0  | 0  | 0  |
+        +----++----+----+----+----+----+----+----+----+----+----+
+        | b4 || b4 | 0  | 0  | 0  | b0 | 0  | 0  | 0  | 0  | 0  |
+        +----++----+----+----+----+----+----+----+----+----+----+
+        | b5 || b5 | 0  | 0  | 0  | 0  | b0 | 0  | 0  | 0  | 0  |
+        +----++----+----+----+----+----+----+----+----+----+----+
+        | b6 || b6 | 0  | 0  | 0  | 0  | 0  | b0 | 0  | 0  | 0  |
+        +----++----+----+----+----+----+----+----+----+----+----+
+        | b7 || b7 | 0  | 0  | 0  | 0  | 0  | 0  | b0 | 0  | 0  |
+        +----++----+----+----+----+----+----+----+----+----+----+
+        | b8 || b8 | 0  | 0  | 0  | 0  | 0  | 0  | 0  | b0 | 0  |
+        +----++----+----+----+----+----+----+----+----+----+----+
+        | b9 || b9 | 0  | 0  | 0  | 0  | 0  | 0  | 0  | 0  | b0 |
+        +----++----+----+----+----+----+----+----+----+----+----+
+
+    TESTS:
+
+    We can actually construct the 27-dimensional Albert algebra,
+    and we get the right unit element if we recompute it::
+
+        sage: J = OctonionHermitianEJA(3,                    # long time
+        ....:                          field=QQ,             # long time
+        ....:                          orthonormalize=False) # long time
+        sage: J.one.clear_cache()                            # long time
+        sage: J.one()                                        # long time
+        b0 + b9 + b26
+        sage: J.one().to_matrix()                            # long time
+        +----+----+----+
+        | e0 | 0  | 0  |
+        +----+----+----+
+        | 0  | e0 | 0  |
+        +----+----+----+
+        | 0  | 0  | e0 |
+        +----+----+----+
+
+    The 2-by-2 algebra is isomorphic to the ten-dimensional Jordan
+    spin algebra, but just to be sure, we recompute its rank::
+
+        sage: J = OctonionHermitianEJA(2,                    # long time
+        ....:                          field=QQ,             # long time
+        ....:                          orthonormalize=False) # long time
+        sage: J.rank.clear_cache()                           # long time
+        sage: J.rank()                                       # long time
+        2
 
-class HadamardEJA(ConcreteEJA):
+    """
+    @staticmethod
+    def _max_random_instance_size():
+        r"""
+        The maximum rank of a random QuaternionHermitianEJA.
+        """
+        return 1 # Dimension 1
+
+    @classmethod
+    def random_instance(cls, **kwargs):
+        """
+        Return a random instance of this type of algebra.
+        """
+        n = ZZ.random_element(cls._max_random_instance_size() + 1)
+        return cls(n, **kwargs)
+
+    def __init__(self, n, field=AA, **kwargs):
+        if n > 3:
+            # Otherwise we don't get an EJA.
+            raise ValueError("n cannot exceed 3")
+
+        # We know this is a valid EJA, but will double-check
+        # if the user passes check_axioms=True.
+        if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
+
+        super().__init__(self._denormalized_basis(n,field),
+                         self.jordan_product,
+                         self.trace_inner_product,
+                         field=field,
+                         **kwargs)
+
+        # TODO: this could be factored out somehow, but is left here
+        # because the MatrixEJA is not presently a subclass of the
+        # FDEJA class that defines rank() and one().
+        self.rank.set_cache(n)
+        idV = self.matrix_space().one()
+        self.one.set_cache(self(idV))
+
+
+    @classmethod
+    def _denormalized_basis(cls, n, field):
+        """
+        Returns a basis for the space of octonion Hermitian n-by-n
+        matrices.
+
+        SETUP::
+
+            sage: from mjo.eja.eja_algebra import OctonionHermitianEJA
+
+        EXAMPLES::
+
+            sage: B = OctonionHermitianEJA._denormalized_basis(3,QQ)
+            sage: all( M.is_hermitian() for M in B )
+            True
+            sage: len(B)
+            27
+
+        """
+        from mjo.octonions import OctonionMatrixAlgebra
+        MS = OctonionMatrixAlgebra(n, scalars=field)
+        es = MS.entry_algebra().gens()
+
+        basis = []
+        for i in range(n):
+            for j in range(i+1):
+                if i == j:
+                    E_ii = MS.monomial( (i,j,es[0]) )
+                    basis.append(E_ii)
+                else:
+                    for e in es:
+                        E_ij  = MS.monomial( (i,j,e)             )
+                        ec = e.conjugate()
+                        # If the conjugate has a negative sign in front
+                        # of it, (j,i,ec) won't be a monomial!
+                        if (j,i,ec) in MS.indices():
+                            E_ij += MS.monomial( (j,i,ec) )
+                        else:
+                            E_ij -= MS.monomial( (j,i,-ec) )
+                        basis.append(E_ij)
+
+        return tuple( basis )
+
+    @staticmethod
+    def trace_inner_product(X,Y):
+        r"""
+        The octonions don't know that the reals are embedded in them,
+        so we have to take the e0 component ourselves.
+
+        SETUP::
+
+            sage: from mjo.eja.eja_algebra import OctonionHermitianEJA
+
+        TESTS::
+
+            sage: J = OctonionHermitianEJA(2,field=QQ,orthonormalize=False)
+            sage: I = J.one().to_matrix()
+            sage: J.trace_inner_product(I, -I)
+            -2
+
+        """
+        return (X*Y).trace().real().coefficient(0)
+
+
+class AlbertEJA(OctonionHermitianEJA):
+    r"""
+    The Albert algebra is the algebra of three-by-three Hermitian
+    matrices whose entries are octonions.
+
+    SETUP::
+
+        from mjo.eja.eja_algebra import AlbertEJA
+
+    EXAMPLES::
+
+        sage: AlbertEJA(field=QQ, orthonormalize=False)
+        Euclidean Jordan algebra of dimension 27 over Rational Field
+        sage: AlbertEJA() # long time
+        Euclidean Jordan algebra of dimension 27 over Algebraic Real Field
+
+    """
+    def __init__(self, *args, **kwargs):
+        super().__init__(3, *args, **kwargs)
+
+
+class HadamardEJA(ConcreteEJA, RationalBasisEJA):
     """
     Return the Euclidean Jordan Algebra corresponding to the set
     `R^n` under the Hadamard product.
@@ -2199,19 +2812,19 @@ class HadamardEJA(ConcreteEJA):
     This multiplication table can be verified by hand::
 
         sage: J = HadamardEJA(3)
-        sage: e0,e1,e2 = J.gens()
-        sage: e0*e0
-        e0
-        sage: e0*e1
+        sage: b0,b1,b2 = J.gens()
+        sage: b0*b0
+        b0
+        sage: b0*b1
         0
-        sage: e0*e2
+        sage: b0*b2
         0
-        sage: e1*e1
-        e1
-        sage: e1*e2
+        sage: b1*b1
+        b1
+        sage: b1*b2
         0
-        sage: e2*e2
-        e2
+        sage: b2*b2
+        b2
 
     TESTS:
 
@@ -2221,12 +2834,17 @@ class HadamardEJA(ConcreteEJA):
         (r0, r1, r2)
 
     """
-    def __init__(self, n, **kwargs):
-        def jordan_product(x,y):
-            P = x.parent()
-            return P(tuple( xi*yi for (xi,yi) in zip(x,y) ))
-        def inner_product(x,y):
-            return x.inner_product(y)
+    def __init__(self, n, field=AA, **kwargs):
+        if n == 0:
+            jordan_product = lambda x,y: x
+            inner_product = lambda x,y: x
+        else:
+            def jordan_product(x,y):
+                P = x.parent()
+                return P( xi*yi for (xi,yi) in zip(x,y) )
+
+            def inner_product(x,y):
+                return (x.T*y)[0,0]
 
         # New defaults for keyword arguments. Don't orthonormalize
         # because our basis is already orthonormal with respect to our
@@ -2237,12 +2855,14 @@ class HadamardEJA(ConcreteEJA):
         if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False
         if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
 
-
-        standard_basis = FreeModule(ZZ, n).basis()
-        super(HadamardEJA, self).__init__(standard_basis,
-                                          jordan_product,
-                                          inner_product,
-                                          **kwargs)
+        column_basis = tuple( b.column()
+                              for b in FreeModule(field, n).basis() )
+        super().__init__(column_basis,
+                         jordan_product,
+                         inner_product,
+                         field=field,
+                         associative=True,
+                         **kwargs)
         self.rank.set_cache(n)
 
         if n == 0:
@@ -2266,7 +2886,7 @@ class HadamardEJA(ConcreteEJA):
         return cls(n, **kwargs)
 
 
-class BilinearFormEJA(ConcreteEJA):
+class BilinearFormEJA(ConcreteEJA, RationalBasisEJA):
     r"""
     The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
     with the half-trace inner product and jordan product ``x*y =
@@ -2346,34 +2966,50 @@ class BilinearFormEJA(ConcreteEJA):
         ....:              for j in range(n-1) ]
         sage: actual == expected
         True
+
     """
-    def __init__(self, B, **kwargs):
-        if not B.is_positive_definite():
-            raise ValueError("bilinear form is not positive-definite")
+    def __init__(self, B, field=AA, **kwargs):
+        # The matrix "B" is supplied by the user in most cases,
+        # so it makes sense to check whether or not its positive-
+        # definite unless we are specifically asked not to...
+        if ("check_axioms" not in kwargs) or kwargs["check_axioms"]:
+            if not B.is_positive_definite():
+                raise ValueError("bilinear form is not positive-definite")
+
+        # However, all of the other data for this EJA is computed
+        # by us in manner that guarantees the axioms are
+        # satisfied. So, again, unless we are specifically asked to
+        # verify things, we'll skip the rest of the checks.
+        if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
 
         def inner_product(x,y):
-            return (B*x).inner_product(y)
+            return (y.T*B*x)[0,0]
 
         def jordan_product(x,y):
             P = x.parent()
-            x0 = x[0]
-            xbar = x[1:]
-            y0 = y[0]
-            ybar = y[1:]
-            z0 = inner_product(x,y)
+            x0 = x[0,0]
+            xbar = x[1:,0]
+            y0 = y[0,0]
+            ybar = y[1:,0]
+            z0 = inner_product(y,x)
             zbar = y0*xbar + x0*ybar
-            return P((z0,) + tuple(zbar))
-
-        # We know this is a valid EJA, but will double-check
-        # if the user passes check_axioms=True.
-        if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
+            return P([z0] + zbar.list())
 
         n = B.nrows()
-        standard_basis = FreeModule(ZZ, n).basis()
-        super(BilinearFormEJA, self).__init__(standard_basis,
-                                              jordan_product,
-                                              inner_product,
-                                              **kwargs)
+        column_basis = tuple( b.column()
+                              for b in FreeModule(field, n).basis() )
+
+        # TODO: I haven't actually checked this, but it seems legit.
+        associative = False
+        if n <= 2:
+            associative = True
+
+        super().__init__(column_basis,
+                         jordan_product,
+                         inner_product,
+                         field=field,
+                         associative=associative,
+                         **kwargs)
 
         # The rank of this algebra is two, unless we're in a
         # one-dimensional ambient space (because the rank is bounded
@@ -2433,20 +3069,20 @@ class JordanSpinEJA(BilinearFormEJA):
     This multiplication table can be verified by hand::
 
         sage: J = JordanSpinEJA(4)
-        sage: e0,e1,e2,e3 = J.gens()
-        sage: e0*e0
-        e0
-        sage: e0*e1
-        e1
-        sage: e0*e2
-        e2
-        sage: e0*e3
-        e3
-        sage: e1*e2
+        sage: b0,b1,b2,b3 = J.gens()
+        sage: b0*b0
+        b0
+        sage: b0*b1
+        b1
+        sage: b0*b2
+        b2
+        sage: b0*b3
+        b3
+        sage: b1*b2
         0
-        sage: e1*e3
+        sage: b1*b3
         0
-        sage: e2*e3
+        sage: b2*b3
         0
 
     We can change the generator prefix::
@@ -2467,7 +3103,7 @@ class JordanSpinEJA(BilinearFormEJA):
             True
 
     """
-    def __init__(self, n, **kwargs):
+    def __init__(self, n, *args, **kwargs):
         # This is a special case of the BilinearFormEJA with the
         # identity matrix as its bilinear form.
         B = matrix.identity(ZZ, n)
@@ -2478,7 +3114,7 @@ class JordanSpinEJA(BilinearFormEJA):
 
         # But also don't pass check_field=False here, because the user
         # can pass in a field!
-        super(JordanSpinEJA, self).__init__(B, **kwargs)
+        super().__init__(B, *args, **kwargs)
 
     @staticmethod
     def _max_random_instance_size():
@@ -2498,7 +3134,7 @@ class JordanSpinEJA(BilinearFormEJA):
         return cls(n, **kwargs)
 
 
-class TrivialEJA(ConcreteEJA):
+class TrivialEJA(ConcreteEJA, RationalBasisEJA):
     """
     The trivial Euclidean Jordan algebra consisting of only a zero element.
 
@@ -2536,10 +3172,12 @@ class TrivialEJA(ConcreteEJA):
         if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False
         if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
 
-        super(TrivialEJA, self).__init__(basis,
-                                         jordan_product,
-                                         inner_product,
-                                         **kwargs)
+        super().__init__(basis,
+                         jordan_product,
+                         inner_product,
+                         associative=True,
+                         **kwargs)
+
         # The rank is zero using my definition, namely the dimension of the
         # largest subalgebra generated by any element.
         self.rank.set_cache(0)
@@ -2551,155 +3189,429 @@ class TrivialEJA(ConcreteEJA):
         # inappropriate for us.
         return cls(**kwargs)
 
-class DirectSumEJA(ConcreteEJA):
+
+class CartesianProductEJA(FiniteDimensionalEJA):
     r"""
-    The external (orthogonal) direct sum of two other Euclidean Jordan
-    algebras. Essentially the Cartesian product of its two factors.
-    Every Euclidean Jordan algebra decomposes into an orthogonal
-    direct sum of simple Euclidean Jordan algebras, so no generality
-    is lost by providing only this construction.
+    The external (orthogonal) direct sum of two or more Euclidean
+    Jordan algebras. Every Euclidean Jordan algebra decomposes into an
+    orthogonal direct sum of simple Euclidean Jordan algebras which is
+    then isometric to a Cartesian product, so no generality is lost by
+    providing only this construction.
 
     SETUP::
 
         sage: from mjo.eja.eja_algebra import (random_eja,
+        ....:                                  CartesianProductEJA,
         ....:                                  HadamardEJA,
-        ....:                                  RealSymmetricEJA,
-        ....:                                  DirectSumEJA)
+        ....:                                  JordanSpinEJA,
+        ....:                                  RealSymmetricEJA)
 
-    EXAMPLES::
+    EXAMPLES:
 
+    The Jordan product is inherited from our factors and implemented by
+    our CombinatorialFreeModule Cartesian product superclass::
+
+        sage: set_random_seed()
         sage: J1 = HadamardEJA(2)
-        sage: J2 = RealSymmetricEJA(3)
-        sage: J = DirectSumEJA(J1,J2)
-        sage: J.dimension()
-        8
+        sage: J2 = RealSymmetricEJA(2)
+        sage: J = cartesian_product([J1,J2])
+        sage: x,y = J.random_elements(2)
+        sage: x*y in J
+        True
+
+    The ability to retrieve the original factors is implemented by our
+    CombinatorialFreeModule Cartesian product superclass::
+
+        sage: J1 = HadamardEJA(2, field=QQ)
+        sage: J2 = JordanSpinEJA(3, field=QQ)
+        sage: J = cartesian_product([J1,J2])
+        sage: J.cartesian_factors()
+        (Euclidean Jordan algebra of dimension 2 over Rational Field,
+         Euclidean Jordan algebra of dimension 3 over Rational Field)
+
+    You can provide more than two factors::
+
+        sage: J1 = HadamardEJA(2)
+        sage: J2 = JordanSpinEJA(3)
+        sage: J3 = RealSymmetricEJA(3)
+        sage: cartesian_product([J1,J2,J3])
+        Euclidean Jordan algebra of dimension 2 over Algebraic Real
+        Field (+) Euclidean Jordan algebra of dimension 3 over Algebraic
+        Real Field (+) Euclidean Jordan algebra of dimension 6 over
+        Algebraic Real Field
+
+    Rank is additive on a Cartesian product::
+
+        sage: J1 = HadamardEJA(1)
+        sage: J2 = RealSymmetricEJA(2)
+        sage: J = cartesian_product([J1,J2])
+        sage: J1.rank.clear_cache()
+        sage: J2.rank.clear_cache()
+        sage: J.rank.clear_cache()
         sage: J.rank()
-        5
+        3
+        sage: J.rank() == J1.rank() + J2.rank()
+        True
+
+    The same rank computation works over the rationals, with whatever
+    basis you like::
+
+        sage: J1 = HadamardEJA(1, field=QQ, orthonormalize=False)
+        sage: J2 = RealSymmetricEJA(2, field=QQ, orthonormalize=False)
+        sage: J = cartesian_product([J1,J2])
+        sage: J1.rank.clear_cache()
+        sage: J2.rank.clear_cache()
+        sage: J.rank.clear_cache()
+        sage: J.rank()
+        3
+        sage: J.rank() == J1.rank() + J2.rank()
+        True
+
+    The product algebra will be associative if and only if all of its
+    components are associative::
+
+        sage: J1 = HadamardEJA(2)
+        sage: J1.is_associative()
+        True
+        sage: J2 = HadamardEJA(3)
+        sage: J2.is_associative()
+        True
+        sage: J3 = RealSymmetricEJA(3)
+        sage: J3.is_associative()
+        False
+        sage: CP1 = cartesian_product([J1,J2])
+        sage: CP1.is_associative()
+        True
+        sage: CP2 = cartesian_product([J1,J3])
+        sage: CP2.is_associative()
+        False
+
+    Cartesian products of Cartesian products work::
+
+        sage: J1 = JordanSpinEJA(1)
+        sage: J2 = JordanSpinEJA(1)
+        sage: J3 = JordanSpinEJA(1)
+        sage: J = cartesian_product([J1,cartesian_product([J2,J3])])
+        sage: J.multiplication_table()
+        +----++----+----+----+
+        | *  || b0 | b1 | b2 |
+        +====++====+====+====+
+        | b0 || b0 | 0  | 0  |
+        +----++----+----+----+
+        | b1 || 0  | b1 | 0  |
+        +----++----+----+----+
+        | b2 || 0  | 0  | b2 |
+        +----++----+----+----+
+        sage: HadamardEJA(3).multiplication_table()
+        +----++----+----+----+
+        | *  || b0 | b1 | b2 |
+        +====++====+====+====+
+        | b0 || b0 | 0  | 0  |
+        +----++----+----+----+
+        | b1 || 0  | b1 | 0  |
+        +----++----+----+----+
+        | b2 || 0  | 0  | b2 |
+        +----++----+----+----+
 
     TESTS:
 
-    The external direct sum construction is only valid when the two factors
-    have the same base ring; an error is raised otherwise::
+    All factors must share the same base field::
 
-        sage: set_random_seed()
-        sage: J1 = random_eja(field=AA)
-        sage: J2 = random_eja(field=QQ,orthonormalize=False)
-        sage: J = DirectSumEJA(J1,J2)
+        sage: J1 = HadamardEJA(2, field=QQ)
+        sage: J2 = RealSymmetricEJA(2)
+        sage: CartesianProductEJA((J1,J2))
         Traceback (most recent call last):
         ...
-        ValueError: algebras must share the same base field
+        ValueError: all factors must share the same base field
+
+    The cached unit element is the same one that would be computed::
+
+        sage: set_random_seed()              # long time
+        sage: J1 = random_eja()              # long time
+        sage: J2 = random_eja()              # long time
+        sage: J = cartesian_product([J1,J2]) # long time
+        sage: actual = J.one()               # long time
+        sage: J.one.clear_cache()            # long time
+        sage: expected = J.one()             # long time
+        sage: actual == expected             # long time
+        True
 
     """
-    def __init__(self, J1, J2, **kwargs):
-        if J1.base_ring() != J2.base_ring():
-            raise ValueError("algebras must share the same base field")
-        field = J1.base_ring()
-
-        self._factors = (J1, J2)
-        n1 = J1.dimension()
-        n2 = J2.dimension()
-        n = n1+n2
-        V = VectorSpace(field, n)
-        mult_table = [ [ V.zero() for j in range(i+1) ]
-                       for i in range(n) ]
-        for i in range(n1):
-            for j in range(i+1):
-                p = (J1.monomial(i)*J1.monomial(j)).to_vector()
-                mult_table[i][j] = V(p.list() + [field.zero()]*n2)
+    Element = FiniteDimensionalEJAElement
 
-        for i in range(n2):
-            for j in range(i+1):
-                p = (J2.monomial(i)*J2.monomial(j)).to_vector()
-                mult_table[n1+i][n1+j] = V([field.zero()]*n1 + p.list())
-
-        # TODO: build the IP table here from the two constituent IP
-        # matrices (it'll be block diagonal, I think).
-        ip_table = [ [ field.zero() for j in range(i+1) ]
-                       for i in range(n) ]
-        super(DirectSumEJA, self).__init__(field,
-                                           mult_table,
-                                           ip_table,
-                                           check_axioms=False,
-                                           **kwargs)
-        self.rank.set_cache(J1.rank() + J2.rank())
-
-
-    def factors(self):
+
+    def __init__(self, factors, **kwargs):
+        m = len(factors)
+        if m == 0:
+            return TrivialEJA()
+
+        self._sets = factors
+
+        field = factors[0].base_ring()
+        if not all( J.base_ring() == field for J in factors ):
+            raise ValueError("all factors must share the same base field")
+
+        associative = all( f.is_associative() for f in factors )
+
+        MS = self.matrix_space()
+        basis = []
+        zero = MS.zero()
+        for i in range(m):
+            for b in factors[i].matrix_basis():
+                z = list(zero)
+                z[i] = b
+                basis.append(z)
+
+        basis = tuple( MS(b) for b in basis )
+
+        # Define jordan/inner products that operate on that matrix_basis.
+        def jordan_product(x,y):
+            return MS(tuple(
+                (factors[i](x[i])*factors[i](y[i])).to_matrix()
+                for i in range(m)
+            ))
+
+        def inner_product(x, y):
+            return sum(
+                factors[i](x[i]).inner_product(factors[i](y[i]))
+                for i in range(m)
+            )
+
+        # There's no need to check the field since it already came
+        # from an EJA. Likewise the axioms are guaranteed to be
+        # satisfied, unless the guy writing this class sucks.
+        #
+        # If you want the basis to be orthonormalized, orthonormalize
+        # the factors.
+        FiniteDimensionalEJA.__init__(self,
+                                      basis,
+                                      jordan_product,
+                                      inner_product,
+                                      field=field,
+                                      orthonormalize=False,
+                                      associative=associative,
+                                      cartesian_product=True,
+                                      check_field=False,
+                                      check_axioms=False)
+
+        ones = tuple(J.one().to_matrix() for J in factors)
+        self.one.set_cache(self(ones))
+        self.rank.set_cache(sum(J.rank() for J in factors))
+
+    def cartesian_factors(self):
+        # Copy/pasted from CombinatorialFreeModule_CartesianProduct.
+        return self._sets
+
+    def cartesian_factor(self, i):
+        r"""
+        Return the ``i``th factor of this algebra.
+        """
+        return self._sets[i]
+
+    def _repr_(self):
+        # Copy/pasted from CombinatorialFreeModule_CartesianProduct.
+        from sage.categories.cartesian_product import cartesian_product
+        return cartesian_product.symbol.join("%s" % factor
+                                             for factor in self._sets)
+
+    def matrix_space(self):
         r"""
-        Return the pair of this algebra's factors.
+        Return the space that our matrix basis lives in as a Cartesian
+        product.
+
+        We don't simply use the ``cartesian_product()`` functor here
+        because it acts differently on SageMath MatrixSpaces and our
+        custom MatrixAlgebras, which are CombinatorialFreeModules. We
+        always want the result to be represented (and indexed) as
+        an ordered tuple.
 
         SETUP::
 
-            sage: from mjo.eja.eja_algebra import (HadamardEJA,
-            ....:                                  JordanSpinEJA,
-            ....:                                  DirectSumEJA)
+            sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+            ....:                                  HadamardEJA,
+            ....:                                  OctonionHermitianEJA,
+            ....:                                  RealSymmetricEJA)
 
         EXAMPLES::
 
-            sage: J1 = HadamardEJA(2, field=QQ)
-            sage: J2 = JordanSpinEJA(3, field=QQ)
-            sage: J = DirectSumEJA(J1,J2)
-            sage: J.factors()
-            (Euclidean Jordan algebra of dimension 2 over Rational Field,
-             Euclidean Jordan algebra of dimension 3 over Rational Field)
+            sage: J1 = HadamardEJA(1)
+            sage: J2 = RealSymmetricEJA(2)
+            sage: J = cartesian_product([J1,J2])
+            sage: J.matrix_space()
+            The Cartesian product of (Full MatrixSpace of 1 by 1 dense
+            matrices over Algebraic Real Field, Full MatrixSpace of 2
+            by 2 dense matrices over Algebraic Real Field)
+
+        ::
+
+            sage: J1 = ComplexHermitianEJA(1)
+            sage: J2 = ComplexHermitianEJA(1)
+            sage: J = cartesian_product([J1,J2])
+            sage: J.one().to_matrix()[0]
+            [1 0]
+            [0 1]
+            sage: J.one().to_matrix()[1]
+            [1 0]
+            [0 1]
+
+        ::
+
+            sage: J1 = OctonionHermitianEJA(1)
+            sage: J2 = OctonionHermitianEJA(1)
+            sage: J = cartesian_product([J1,J2])
+            sage: J.one().to_matrix()[0]
+            +----+
+            | e0 |
+            +----+
+            sage: J.one().to_matrix()[1]
+            +----+
+            | e0 |
+            +----+
 
         """
-        return self._factors
+        scalars = self.cartesian_factor(0).base_ring()
+
+        # This category isn't perfect, but is good enough for what we
+        # need to do.
+        cat = MagmaticAlgebras(scalars).FiniteDimensional().WithBasis()
+        cat = cat.Unital().CartesianProducts()
+        factors = tuple( J.matrix_space() for J in self.cartesian_factors() )
+
+        from sage.sets.cartesian_product import CartesianProduct
+        return CartesianProduct(factors, cat)
 
-    def projections(self):
-        r"""
-        Return a pair of projections onto this algebra's factors.
 
+    @cached_method
+    def cartesian_projection(self, i):
+        r"""
         SETUP::
 
-            sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
-            ....:                                  ComplexHermitianEJA,
-            ....:                                  DirectSumEJA)
+            sage: from mjo.eja.eja_algebra import (random_eja,
+            ....:                                  JordanSpinEJA,
+            ....:                                  HadamardEJA,
+            ....:                                  RealSymmetricEJA,
+            ....:                                  ComplexHermitianEJA)
 
-        EXAMPLES::
+        EXAMPLES:
+
+        The projection morphisms are Euclidean Jordan algebra
+        operators::
+
+            sage: J1 = HadamardEJA(2)
+            sage: J2 = RealSymmetricEJA(2)
+            sage: J = cartesian_product([J1,J2])
+            sage: J.cartesian_projection(0)
+            Linear operator between finite-dimensional Euclidean Jordan
+            algebras represented by the matrix:
+            [1 0 0 0 0]
+            [0 1 0 0 0]
+            Domain: Euclidean Jordan algebra of dimension 2 over Algebraic
+            Real Field (+) Euclidean Jordan algebra of dimension 3 over
+            Algebraic Real Field
+            Codomain: Euclidean Jordan algebra of dimension 2 over Algebraic
+            Real Field
+            sage: J.cartesian_projection(1)
+            Linear operator between finite-dimensional Euclidean Jordan
+            algebras represented by the matrix:
+            [0 0 1 0 0]
+            [0 0 0 1 0]
+            [0 0 0 0 1]
+            Domain: Euclidean Jordan algebra of dimension 2 over Algebraic
+            Real Field (+) Euclidean Jordan algebra of dimension 3 over
+            Algebraic Real Field
+            Codomain: Euclidean Jordan algebra of dimension 3 over Algebraic
+            Real Field
+
+        The projections work the way you'd expect on the vector
+        representation of an element::
 
             sage: J1 = JordanSpinEJA(2)
             sage: J2 = ComplexHermitianEJA(2)
-            sage: J = DirectSumEJA(J1,J2)
-            sage: (pi_left, pi_right) = J.projections()
-            sage: J.one().to_vector()
-            (1, 0, 1, 0, 0, 1)
+            sage: J = cartesian_product([J1,J2])
+            sage: pi_left = J.cartesian_projection(0)
+            sage: pi_right = J.cartesian_projection(1)
             sage: pi_left(J.one()).to_vector()
             (1, 0)
             sage: pi_right(J.one()).to_vector()
             (1, 0, 0, 1)
+            sage: J.one().to_vector()
+            (1, 0, 1, 0, 0, 1)
+
+        TESTS:
+
+        The answer never changes::
+
+            sage: set_random_seed()
+            sage: J1 = random_eja()
+            sage: J2 = random_eja()
+            sage: J = cartesian_product([J1,J2])
+            sage: P0 = J.cartesian_projection(0)
+            sage: P1 = J.cartesian_projection(0)
+            sage: P0 == P1
+            True
 
         """
-        (J1,J2) = self.factors()
-        m = J1.dimension()
-        n = J2.dimension()
-        V_basis = self.vector_space().basis()
-        # Need to specify the dimensions explicitly so that we don't
-        # wind up with a zero-by-zero matrix when we want e.g. a
-        # zero-by-two matrix (important for composing things).
-        P1 = matrix(self.base_ring(), m, m+n, V_basis[:m])
-        P2 = matrix(self.base_ring(), n, m+n, V_basis[m:])
-        pi_left = FiniteDimensionalEJAOperator(self,J1,P1)
-        pi_right = FiniteDimensionalEJAOperator(self,J2,P2)
-        return (pi_left, pi_right)
-
-    def inclusions(self):
-        r"""
-        Return the pair of inclusion maps from our factors into us.
+        offset = sum( self.cartesian_factor(k).dimension()
+                      for k in range(i) )
+        Ji = self.cartesian_factor(i)
+        Pi = self._module_morphism(lambda j: Ji.monomial(j - offset),
+                                   codomain=Ji)
 
+        return FiniteDimensionalEJAOperator(self,Ji,Pi.matrix())
+
+    @cached_method
+    def cartesian_embedding(self, i):
+        r"""
         SETUP::
 
             sage: from mjo.eja.eja_algebra import (random_eja,
             ....:                                  JordanSpinEJA,
-            ....:                                  RealSymmetricEJA,
-            ....:                                  DirectSumEJA)
+            ....:                                  HadamardEJA,
+            ....:                                  RealSymmetricEJA)
 
-        EXAMPLES::
+        EXAMPLES:
+
+        The embedding morphisms are Euclidean Jordan algebra
+        operators::
+
+            sage: J1 = HadamardEJA(2)
+            sage: J2 = RealSymmetricEJA(2)
+            sage: J = cartesian_product([J1,J2])
+            sage: J.cartesian_embedding(0)
+            Linear operator between finite-dimensional Euclidean Jordan
+            algebras represented by the matrix:
+            [1 0]
+            [0 1]
+            [0 0]
+            [0 0]
+            [0 0]
+            Domain: Euclidean Jordan algebra of dimension 2 over
+            Algebraic Real Field
+            Codomain: Euclidean Jordan algebra of dimension 2 over
+            Algebraic Real Field (+) Euclidean Jordan algebra of
+            dimension 3 over Algebraic Real Field
+            sage: J.cartesian_embedding(1)
+            Linear operator between finite-dimensional Euclidean Jordan
+            algebras represented by the matrix:
+            [0 0 0]
+            [0 0 0]
+            [1 0 0]
+            [0 1 0]
+            [0 0 1]
+            Domain: Euclidean Jordan algebra of dimension 3 over
+            Algebraic Real Field
+            Codomain: Euclidean Jordan algebra of dimension 2 over
+            Algebraic Real Field (+) Euclidean Jordan algebra of
+            dimension 3 over Algebraic Real Field
+
+        The embeddings work the way you'd expect on the vector
+        representation of an element::
 
             sage: J1 = JordanSpinEJA(3)
             sage: J2 = RealSymmetricEJA(2)
-            sage: J = DirectSumEJA(J1,J2)
-            sage: (iota_left, iota_right) = J.inclusions()
+            sage: J = cartesian_product([J1,J2])
+            sage: iota_left = J.cartesian_embedding(0)
+            sage: iota_right = J.cartesian_embedding(1)
             sage: iota_left(J1.zero()) == J.zero()
             True
             sage: iota_right(J2.zero()) == J.zero()
@@ -2717,6 +3629,17 @@ class DirectSumEJA(ConcreteEJA):
 
         TESTS:
 
+        The answer never changes::
+
+            sage: set_random_seed()
+            sage: J1 = random_eja()
+            sage: J2 = random_eja()
+            sage: J = cartesian_product([J1,J2])
+            sage: E0 = J.cartesian_embedding(0)
+            sage: E1 = J.cartesian_embedding(0)
+            sage: E0 == E1
+            True
+
         Composing a projection with the corresponding inclusion should
         produce the identity map, and mismatching them should produce
         the zero map::
@@ -2724,9 +3647,11 @@ class DirectSumEJA(ConcreteEJA):
             sage: set_random_seed()
             sage: J1 = random_eja()
             sage: J2 = random_eja()
-            sage: J = DirectSumEJA(J1,J2)
-            sage: (iota_left, iota_right) = J.inclusions()
-            sage: (pi_left, pi_right) = J.projections()
+            sage: J = cartesian_product([J1,J2])
+            sage: iota_left = J.cartesian_embedding(0)
+            sage: iota_right = J.cartesian_embedding(1)
+            sage: pi_left = J.cartesian_projection(0)
+            sage: pi_right = J.cartesian_projection(1)
             sage: pi_left*iota_left == J1.one().operator()
             True
             sage: pi_right*iota_right == J2.one().operator()
@@ -2737,58 +3662,83 @@ class DirectSumEJA(ConcreteEJA):
             True
 
         """
-        (J1,J2) = self.factors()
-        m = J1.dimension()
-        n = J2.dimension()
-        V_basis = self.vector_space().basis()
-        # Need to specify the dimensions explicitly so that we don't
-        # wind up with a zero-by-zero matrix when we want e.g. a
-        # two-by-zero matrix (important for composing things).
-        I1 = matrix.column(self.base_ring(), m, m+n, V_basis[:m])
-        I2 = matrix.column(self.base_ring(), n, m+n, V_basis[m:])
-        iota_left = FiniteDimensionalEJAOperator(J1,self,I1)
-        iota_right = FiniteDimensionalEJAOperator(J2,self,I2)
-        return (iota_left, iota_right)
+        offset = sum( self.cartesian_factor(k).dimension()
+                      for k in range(i) )
+        Ji = self.cartesian_factor(i)
+        Ei = Ji._module_morphism(lambda j: self.monomial(j + offset),
+                                 codomain=self)
+        return FiniteDimensionalEJAOperator(Ji,self,Ei.matrix())
 
-    def inner_product(self, x, y):
-        r"""
-        The standard Cartesian inner-product.
 
-        We project ``x`` and ``y`` onto our factors, and add up the
-        inner-products from the subalgebras.
 
-        SETUP::
+FiniteDimensionalEJA.CartesianProduct = CartesianProductEJA
 
+class RationalBasisCartesianProductEJA(CartesianProductEJA,
+                                       RationalBasisEJA):
+    r"""
+    A separate class for products of algebras for which we know a
+    rational basis.
 
-            sage: from mjo.eja.eja_algebra import (HadamardEJA,
-            ....:                                  QuaternionHermitianEJA,
-            ....:                                  DirectSumEJA)
-
-        EXAMPLE::
-
-            sage: J1 = HadamardEJA(3,field=QQ)
-            sage: J2 = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False)
-            sage: J = DirectSumEJA(J1,J2)
-            sage: x1 = J1.one()
-            sage: x2 = x1
-            sage: y1 = J2.one()
-            sage: y2 = y1
-            sage: x1.inner_product(x2)
-            3
-            sage: y1.inner_product(y2)
-            2
-            sage: J.one().inner_product(J.one())
-            5
+    SETUP::
 
-        """
-        (pi_left, pi_right) = self.projections()
-        x1 = pi_left(x)
-        x2 = pi_right(x)
-        y1 = pi_left(y)
-        y2 = pi_right(y)
+        sage: from mjo.eja.eja_algebra import (HadamardEJA,
+        ....:                                  JordanSpinEJA,
+        ....:                                  OctonionHermitianEJA,
+        ....:                                  RealSymmetricEJA)
+
+    EXAMPLES:
 
-        return (x1.inner_product(y1) + x2.inner_product(y2))
+    This gives us fast characteristic polynomial computations in
+    product algebras, too::
 
 
+        sage: J1 = JordanSpinEJA(2)
+        sage: J2 = RealSymmetricEJA(3)
+        sage: J = cartesian_product([J1,J2])
+        sage: J.characteristic_polynomial_of().degree()
+        5
+        sage: J.rank()
+        5
 
-random_eja = ConcreteEJA.random_instance
+    TESTS:
+
+    The ``cartesian_product()`` function only uses the first factor to
+    decide where the result will live; thus we have to be careful to
+    check that all factors do indeed have a `_rational_algebra` member
+    before we try to access it::
+
+        sage: J1 = OctonionHermitianEJA(1) # no rational basis
+        sage: J2 = HadamardEJA(2)
+        sage: cartesian_product([J1,J2])
+        Euclidean Jordan algebra of dimension 1 over Algebraic Real Field
+        (+) Euclidean Jordan algebra of dimension 2 over Algebraic Real Field
+        sage: cartesian_product([J2,J1])
+        Euclidean Jordan algebra of dimension 2 over Algebraic Real Field
+        (+) Euclidean Jordan algebra of dimension 1 over Algebraic Real Field
+
+    """
+    def __init__(self, algebras, **kwargs):
+        CartesianProductEJA.__init__(self, algebras, **kwargs)
+
+        self._rational_algebra = None
+        if self.vector_space().base_field() is not QQ:
+            if all( hasattr(r, "_rational_algebra") for r in algebras ):
+                self._rational_algebra = cartesian_product([
+                    r._rational_algebra for r in algebras
+                ])
+
+
+RationalBasisEJA.CartesianProduct = RationalBasisCartesianProductEJA
+
+def random_eja(*args, **kwargs):
+    J1 = ConcreteEJA.random_instance(*args, **kwargs)
+
+    # This might make Cartesian products appear roughly as often as
+    # any other ConcreteEJA.
+    if ZZ.random_element(len(ConcreteEJA.__subclasses__()) + 1) == 0:
+        # Use random_eja() again so we can get more than two factors.
+        J2 = random_eja(*args, **kwargs)
+        J = cartesian_product([J1,J2])
+        return J
+    else:
+        return J1