1. Add cartesian products to random_eja().

2. Add references and start citing them.

3. Implement the octonion simple EJA. We don't actually need octonions
   for this to work, only their real embedding (some 8x8 monstrosity).

4. Pre-cache charpoly for some small algebras?

RealSymmetricEJA(4):

sage: F = J.base_ring()
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]*X[7] + (F(2).sqrt()/2)*X[1]*X[5]*X[6]*X[7] + (1/4)*X[3]**2*X[7]**2 - (1/2)*X[0]*X[5]*X[7]**2 + (F(2).sqrt()/2)*X[2]*X[3]*X[6]*X[8] - (1/2)*X[1]*X[4]*X[6*X[8] - (1/2)*X[1]*X[3]*X[7]*X[8] + (F(2).sqrt()/2)*X[0]*X[4]*X[7]*X[8] + (1/4)*X[1]**2*X[8]**2 - (1/2)*X[0]*X[2]*X[8]**2 - (1/2)*X[2]*X[3]**2*X[9] + (F(2).sqrt()/2)*X[1]*X[3]*X[4]*X[9] - (1/2)*X[0]*X[4]**2*X[9] - (1/2)*X[1]**2*X[5]*X[9] + X[0]*X[2]*X[5]*X[9]

5. Profile the construction of "large" matrix algebras (like the
   15-dimensional QuaternionHermitianAlgebra(3)) to find out why
   they're so slow.

6. Instead of storing a basis multiplication matrix, just make
   product_on_basis() a cached method and manually cache its
   entries. The cython cached method lookup should be faster than a
   python-based matrix lookup anyway. NOTE: we should still be able
   to recompute the table somehow. Is this worth it?

7. What the ever-loving fuck is this shit?

       sage: O = Octonions(QQ)
       sage: e0 = O.monomial(0)
       sage: e0*[[[[]]]]
       [[[[]]]]*e0

8. In fact, could my octonion matrix algebra be generalized for any
   algebra of matrices over the reals whose entries are not real? Then
   we wouldn't need real embeddings at all. They might even be fricking
   vector spaces if I did that...

9. Add HurwitzMatrixAlgebra subclass between MatrixAlgebra and
   OctonionMatrixAlgebra.