Cube map is endomorphism implies class three

Statement
Suppose $$G$$ is a group such that the cube map $$x \mapsto x^3$$ is an endomorphism of $$G$$.

Then, $$G$$ is a nilpotent group and its nilpotency class is at most three.

Facts used

 * 1) uses::Levi's characterization of 3-abelian groups
 * 2) uses::2-Engel implies class three for groups

Proof
The proof follows directly by combining Facts (1) and (2).