Central and additive group of a commutative unital ring implies potentially iterated commutator subgroup in solvable group

Statement

Suppose $G$ is a solvable group and $H$ is a central subgroup that is isomorphic to the additive group of a commutative unital ring. Then, $H$ is a Potentially iterated commutator subgroup (?) of $G$, i.e., there exists a group $K$ containing $G$ such that $H$ is an iterated commutator subgroup of $K$.

In particular, $H$ is a Potentially verbal subgroup (?) and a Potentially fully invariant subgroup (?) of $G$.

Proof

Given: A solvable group $G$, a central subgroup $H$ of $G$, a commutative unital ring $R$ such that $H$ is isomorphic to the additive group of $R$.

To prove: There exists a group $K$ containing $G$ such that $H$ is an iterated commutator subgroup of $K$.

Proof: Suppose $G$ has derived length $l$. Consider $L = U(2^{l+1} - 1,R)$, i.e., the group of $(2^l - 1) \times (2^l - 1)$ upper-triangular matrices with $1$s in the diagonal and entries from $R$, under multiplication. $L$ has derived length $l + 1$ and its center as well as the $l^{th}$ member of its derived series is a subgroup isomorphic to $H$. Let $K$ be the central product of $G$ and $L$ with the center of $L$ identified with $H$.

Then, the $l^{th}$ member of the derived series of $K$ equals the common subgroup $H$. Thus, $H$ is an iterated commutator subgroup of $K$.