Verbal not implies commutator-verbal

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., verbal subgroup) need not satisfy the second subgroup property (i.e., commutator-verbal subgroup)
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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 $p$.