In a group $G$, if $a^3=e$ for all $a$ belongs to $G$, then is $G$ abelian?

Question

It's easy to show that $G$ is abelian if $a^2=e.$

Can't seem to figure out how to prove/disprove this.

Answer 1

The standard counterexample is the group of matrices of the form $$\pmatrix{1&a&b\\0&1&c\\0&0&1}$$ over the field of three elements.