Centralizer of derived subgroup has class at most two

Statement
Suppose $$G$$ is a group and $$H = C_G([G,G])$$ is the fact about::centralizer of commutator subgroup of $$G$$. Then, $$H$$ is a fact about::group of nilpotency class two. In other words, $$H$$ is either an abelian group or a non-abelian group of class two. In particular, $$H$$ is a fact about::class two normal subgroup and a fact about::class two characteristic subgroup.

Related facts

 * Commutator subgroup centralizes cyclic normal subgroup
 * Commutator subgroup centralizes aut-abelian normal subgroup
 * Centralizer of commutator subgroup is hereditarily 2-subnormal
 * Abelian Frattini subgroup implies centralizer is critical

Proof
Given: A group $$G$$. $$H = C_G([G,G])$$.

To prove: $$H$$ is a group of class (at most) two.

Proof: Since $$[G,G]$$ centralizes $$H$$, and $$H \le G$$, $$[H,H]$$ centralizes $$H$$. Thus, $$[H,H]$$ is contained in the center of $$H$$. Thus, $$H$$ has class at most two.