On a reformulation of the commutator subgroup

For semigroup $S$, a commutative congruence ${\sigma }_{orient}$ on $S$ and a subsemigroup Orientable($S$) of $S$ were introduced in “Two cancellative commutative congruences and group diagrams”, Semigroup Forum (2011) 82: 338-353. Here we demonstrate that when the semigroup is in fact a group $G$, then Orientable($G$) is the commutator subgroup $[G,G]$ and $G\mathrm{/}{\sigma }_{orient}$ is the abelian quotient group $G\mathrm{/}[G,G]$.