> That changes the meaning. You're now relying on multiplication being commutative which is not true for all sets

As I said at the end of my last comment: I don't see a case where the ambiguity would pop up in real discourse. If you are dealing with non-commutative objects you can deal with a few more parenthesis. Just like non-associative objects result in many more.

It doesn't matter in practice. These two bars denote either absolute value, or norm, or determinant, all of which are real or complex numbers. People surely have the freedom to define a non-commutative multiplication with real/complex scalars, but have you ever seen one?