Yeah, I think it's informative that you "need commutativity" but important that you can build it yourself. It's nice (mostly from an efficiency standpoint, sometimes from a complexity stnadpoint) when you can get it "for free" because the underlying type is commutative, and the fact that you're shooting for commutativity can inform how you build and test the bookkeeping.
As an interesting nit, "anticommutative" specifically means that a.b = -(b.a), which is different than simply not being commutative.
A group might be commutative, anticommutative, neither, or even both (trivially true of the empty group and the group with one element, but I think it can be true of larger groups).
Ah, my terminology is rusty as it's been years since my math classes. I guess I need to review my algebra books again. I do have some personal code that called "exists a,b a.b != b.a" anticommutative, didn't even realize that's wrong terminology!
As an interesting nit, "anticommutative" specifically means that a.b = -(b.a), which is different than simply not being commutative.
A group might be commutative, anticommutative, neither, or even both (trivially true of the empty group and the group with one element, but I think it can be true of larger groups).