Last time I had asked for intuitions about associativity, @JasonHise64 suggested that associativity is commutativity squared. In retrospect, his suggestion seems much clearer than I had initially been able to see. But I think I got it now. Let's take a look together... (thread)
There's also an associator [A,B,C] = (AB)C - A(BC) about which I know next to nothing. But I know a little more about commutators. For these the Jacobi identity holds [A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0 which is kind of very similar to associativity when written like this:
[[A,B],C] - [A,[B,C]] = [[A,C],B]
Oh no, now it starts all over again! What's the symmetry group of that, and how can we read it off a multiplication table ...
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