TITLE: Prove this magma commutativity
QUESTION [5 upvotes]: Given magma $(X, *)$ and that $(x*y)*y = x$ and $y*(y*x) = x$ $\forall x, y \in X$ prove that $x*y = y*x$.
Should be rather simple but I've been trying to prove that for several hours now with no luck. I've been trying lots of substitutions in $x*y$ and $y*x$ but wasn't able to come to something identical.

REPLY [2 votes]: In short: $$x*y=((x*y)*x)*x=((x*y)*((x*y)*y))*x=y*x$$