-8. The inner product should be an *argument* to the main EJA
- constructor. Afterwards, the basis normalization step should be
- optional (and enabled by default) for ALL algebras, since any
- algebra can have a nonstandard inner-product and its basis can be
- normalized with respect to that inner- product. For example, the
- HadamardEJA could be equipped with an inner- product that is twice
- the usual one. Then for the basis to be orthonormal, we would need
- to divide e.g. (1,0,0) by <(1,0,0),(1,0,0)> = 2 to normalize 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. 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.