+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.