Verbal not implies commutator-verbal

Statement
it is possible to have a group $$G$$ and a verbal subgroup $$H$$ of $$G$$ that is not a commutator-verbal subgroup of $$G$$.

Proof
Any example where $$G$$ is an abelian group and $$H$$ is a proper nontrivial verbal subgroup of $$G$$ suffices. For instance, $$G$$ is the cyclic group of order $$p^2$$ for some prime $$p$$, and $$H$$ is the set of multiples of $$p$$.