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 fact about::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 fact about::potentially verbal subgroup and a fact about::potentially fully invariant subgroup of $$G$$.

Related facts

 * Central implies finite-pi-potentially verbal in finite
 * Cyclic normal implies finite-pi-potentially verbal in finite
 * Homocyclic normal implies finite-pi-potentially fully invariant in finite
 * Abelian normal subgroup of finite group with induced quotient action by power automorphisms is finite-pi-potentially verbal

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 $$K$$.