Show there exists a non-commutative cancellative semigroup with generators $w,x$ satisfying $xwx=ww$

Question

Let $W$ be a cancellative semigroup and $w,x\in W$ non-identity elements such that $xwx=ww$ and $\{x,w\}$ generates $W$. Prove that $W$ is commutative or find a counterexample. (Note that $x=1,w=2,W=\mathbb{N}$ is a commutative example.)

Answer 1

Answer was found on another website: $S_3$ with $x=(1\ 2)$ and $w=(1\ 2\ 3)$.