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the binary operation should also be associative (e.g. subtraction is no good)

For 2x2 matrix multiplication, it is known that Strassen's algorithm is optimal in the sense that there exists no algebraic circuit that uses only 6 "scalar" multiplications. I took your question to ask whether for 3x3 matrix multiplication, the minimum number of scalar multiplications required was similarly known. It is not, and there are more options to take into account than just orderings.

It seemed to me that the paragraph you quoted does not much illuminate why one should choose for one or the other.

No!

But both class and className are camelCased.