From d6c744ecba0a22fdd76cb17e663594d323d1bb38 Mon Sep 17 00:00:00 2001
From: Michael Orlitzky <michael@orlitzky.com>
Date: Sun, 7 Mar 2021 01:28:45 -0500
Subject: [PATCH] eja: make OctonionHermitianEJAs RationalBasisEJAs.

---
 mjo/eja/TODO           |  8 +++++---
 mjo/eja/eja_algebra.py | 24 ++++++++++++++----------
 2 files changed, 19 insertions(+), 13 deletions(-)

diff --git a/mjo/eja/TODO b/mjo/eja/TODO
index cd516fc..1742285 100644
--- a/mjo/eja/TODO
+++ b/mjo/eja/TODO
@@ -24,9 +24,11 @@ sage: a0 = (1/4)*X[4]**2*X[6]**2 - (1/2)*X[2]*X[5]*X[6]**2 - (1/2)*X[3]*X[4]*X[6
        sage: e0*[[[[]]]]
        [[[[]]]]*e0
 
-6. Figure out how to make OctonionHermitianEJA a RationalBasisEJA,
-   and then convert the complex/quaternion algebras to avoid real-
-   (un)embeddings.
+6. Can we convert the complex/quaternion algebras to avoid real-
+   (un)embeddings? Quaternions would need their own
+   QuaternionMatrixAlgebra, since Sage matrices have to have entries
+   in a commutative ring. Those and the octonion stuff could be moved
+   to hurwitz.py along with the HurwitzMatrixAlgebra.
 
 7. Every once in a long while, the test
 
diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index ca3df5f..6b5b090 100644
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -1575,7 +1575,11 @@ class RationalBasisEJA(FiniteDimensionalEJA):
         if check_field:
             # Abuse the check_field parameter to check that the entries of
             # out basis (in ambient coordinates) are in the field QQ.
-            if not all( all(b_i in QQ for b_i in b.list()) for b in basis ):
+            # Use _all2list to get the vector coordinates of octonion
+            # entries and not the octonions themselves (which are not
+            # rational).
+            if not all( all(b_i in QQ for b_i in _all2list(b))
+                        for b in basis ):
                 raise TypeError("basis not rational")
 
         super().__init__(basis,
@@ -1832,7 +1836,7 @@ class RealEmbeddedMatrixEJA(MatrixEJA):
         # 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, RationalBasisEJA, MatrixEJA):
+class RealSymmetricEJA(RationalBasisEJA, ConcreteEJA, MatrixEJA):
     """
     The rank-n simple EJA consisting of real symmetric n-by-n
     matrices, the usual symmetric Jordan product, and the trace inner
@@ -2114,7 +2118,7 @@ class ComplexMatrixEJA(RealEmbeddedMatrixEJA):
         return matrix(F, n/d, elements)
 
 
-class ComplexHermitianEJA(ConcreteEJA, RationalBasisEJA, ComplexMatrixEJA):
+class ComplexHermitianEJA(RationalBasisEJA, ConcreteEJA, ComplexMatrixEJA):
     """
     The rank-n simple EJA consisting of complex Hermitian n-by-n
     matrices over the real numbers, the usual symmetric Jordan product,
@@ -2416,8 +2420,8 @@ class QuaternionMatrixEJA(RealEmbeddedMatrixEJA):
         return matrix(Q, n/d, elements)
 
 
-class QuaternionHermitianEJA(ConcreteEJA,
-                             RationalBasisEJA,
+class QuaternionHermitianEJA(RationalBasisEJA,
+                             ConcreteEJA,
                              QuaternionMatrixEJA):
     r"""
     The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
@@ -2586,7 +2590,7 @@ class QuaternionHermitianEJA(ConcreteEJA,
         n = ZZ.random_element(cls._max_random_instance_size() + 1)
         return cls(n, **kwargs)
 
-class OctonionHermitianEJA(ConcreteEJA, MatrixEJA):
+class OctonionHermitianEJA(RationalBasisEJA, ConcreteEJA, MatrixEJA):
     r"""
     SETUP::
 
@@ -2780,7 +2784,7 @@ class AlbertEJA(OctonionHermitianEJA):
 
     SETUP::
 
-        from mjo.eja.eja_algebra import AlbertEJA
+        sage: from mjo.eja.eja_algebra import AlbertEJA
 
     EXAMPLES::
 
@@ -2794,7 +2798,7 @@ class AlbertEJA(OctonionHermitianEJA):
         super().__init__(3, *args, **kwargs)
 
 
-class HadamardEJA(ConcreteEJA, RationalBasisEJA):
+class HadamardEJA(RationalBasisEJA, ConcreteEJA):
     """
     Return the Euclidean Jordan Algebra corresponding to the set
     `R^n` under the Hadamard product.
@@ -2886,7 +2890,7 @@ class HadamardEJA(ConcreteEJA, RationalBasisEJA):
         return cls(n, **kwargs)
 
 
-class BilinearFormEJA(ConcreteEJA, RationalBasisEJA):
+class BilinearFormEJA(RationalBasisEJA, ConcreteEJA):
     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 =
@@ -3134,7 +3138,7 @@ class JordanSpinEJA(BilinearFormEJA):
         return cls(n, **kwargs)
 
 
-class TrivialEJA(ConcreteEJA, RationalBasisEJA):
+class TrivialEJA(RationalBasisEJA, ConcreteEJA):
     """
     The trivial Euclidean Jordan algebra consisting of only a zero element.
 
-- 
2.44.2