Class two implies generated by abelian normal subgroups

Statement
Suppose $$P$$ is a group of nilpotency class two. Then, $$P$$ is a group generated by abelian normal subgroups. In fact, $$P$$ is a union of abelian normal subgroups.

Facts used

 * 1) uses::Every group is a union of cyclic subgroups
 * 2) uses::Abelian implies every subgroup is normal
 * 3) uses::Cyclic over central implies abelian
 * 4) uses::Normality satisfies inverse image condition

Proof
Given: A group $$P$$ with center $$C$$ such that $$G := P/C$$ is an abelian group.

To prove: $$P$$ is a union of abelian normal subgroups of itself.

Proof: