subalgebras B and C of A, we can use to_matrix() to go from, say,
C -> A -> B rather than having to convert from C to B directly.
-3. When constructing a Cartesian product algebra, we don't know a
- priori whether or not the result will have matrix-algebra
- factors. We can figure it out at runtime, but it would be nice if
- DirectSumEJA always returned the same class. Maybe more
- importantly, if a Cartesian product has one matrix and one
- non-matrix factor, then what would its own matrix representation
- look like? We want to delegate to the factors...
+Fielding questions
+------------------
+
+All Euclidean Jordan algebras are over the real scalar field. This
+presents a problem: in SageMath, the matrix and vector classes don't
+support scalar fields that are different than their entries. And three
+of the simple families of Euclidean Jordan algebras are matrices with
+non-real entries: the Hermitian comples, quaternion, and octonion
+algebras.
+
+At least in the complex and quaternion case, we can "embed" the
+complex numbers and quaternions into the space of 2-by-2 or 4-by-4
+matrices. But the octonions are not associative, so they can't be
+embedded (via a homomorphism) into any real matrix space. So what
+do we do? Write it ourselves, obviously.
+
+The octonion matrix algebra is implemented separately, as a subclass
+of CombinatorialFreeModule, to work around that issue. The custom
+class supports a scalar field that is different than the entries of
+the matrices. However, this means that we actually have FOUR
+different types of "matrices" to support:
+
+ (1) Sage vectors
+ (2) Sage matrices
+ (3) Our custom matrices
+ (4) Cartesian products of the (1) through (3)
+
+All other Euclidean Jordan algebras could of course be implemented in
+the same way as the octonion algebra, but for the sake of the user
+interface, we must also support at least the usual SageMath vectors
+and matrices.
+
+Note: this has one less-than-obvious consequence: we have to assume
+that the user has supplied an entirely-correct basis (with entries in
+the correct structure). We generally cannot mess witht the entries of
+his basis, or use them to figure out what (for example) the ambient
+scalar ring is. None of these are insurmountable obstacles; we just
+have to be a little careful distinguishing between what's inside the
+algebra elements and what's outside them.
Basis normalization
-------------------
computations in the original basis (like the "characteristic
polynomial of"), which can be much faster when the original basis
contains only rational entries.
+
+Debugging
+---------
+There are several places in the code where we set check_field=False
+and check_axioms=False because the theory guarantees that they will be
+satisfied. Well, you know what they say about theory and practice. The
+first thing you should do when a problem is discovered it replace all
+of those with check_field=True and check_axioms=True, and then re-run
+the test suite.