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. Levi's characterization of 3-abelian groups
2. 2-Engel implies class three for groups

Proof

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