Nilpotent and every abelian characteristic subgroup is central implies class at most two

Statement
If $$G$$ is a fact about::nilpotent group and satisfies the property that every Abelian characteristic subgroup of it is central, then the nilpotence class of $$G$$ is at most two.

Facts used

 * 1) uses::Penultimate term of lower central series is Abelian in nilpotent group of class at least three

Proof
The proof follows direct from fact (1): it shows that in a nilpotent group of class at least three, we have an Abelian characteristic subgroup that is not central (namely, the penultimate term of the lower central series).