Additive group of a commutative unital ring implies commutator-realizable

Statement
Suppose $$G$$ is an abelian group that is isomorphic to the additive group of a commutative unital ring. Then, $$G$$ is a commutator-realizable group. In other words, there exists a group $$K$$ containing $$G$$ such that the commutator subgroup of $$K$$ equals $$G$$.

Proof
Given: An abelian group $$G$$ that is the additive group of a commutative unital ring $$R$$.

To prove: There exists a group $$K$$ such that the commutator subgroup $$[K,K]$$ equals $$G$$.

Proof: Consider the group $$U(3,R)$$ of upper-triangular $$3 \times 3$$ matrices with $$1$$s on the diagonal under multiplication. $$U(3,R)$$ is a group of nilpotency class two, whose commutator subgroup and cneter are both the group of matrices of the form:

$$\begin{pmatrix} 1 & 0 & a \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix}$$

which is isomorphic to the additive group $$G$$ of $$R$$.