-All other Euclidean Jordan algebras could of course be implemented in
-the same way as the octonion algebra, but for the sake of the user
-interface, we must also support at least the usual SageMath vectors
-and matrices.
+The real symmetric matrices could of course be implemented in the same
+manner as the others; but for the sake of the user interface, we must
+also support at least the usual SageMath vectors and matrices. Having
+the real symmetric matrices actually be (SageMath) matrices ensures
+that we don't accidentally break support for such things.
+
+Note: this has one less-than-obvious consequence: we have to assume
+that the user has supplied an entirely-correct basis (with entries in
+the correct structure). We generally cannot mess witht the entries of
+his basis, or use them to figure out what (for example) the ambient
+scalar ring is. None of these are insurmountable obstacles; we just
+have to be a little careful distinguishing between what's inside the
+algebra elements and what's outside them.