1. Add references and start citing them.

2. 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]

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

4. 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?

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

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

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.

7. Every once in a long while, the test

       sage: set_random_seed()
       sage: x = random_eja().random_element()
       sage: x.is_invertible() == (x.det() != 0)

   in eja_element.py returns False.