Levi's characterization of 3-abelian groups

Statement
The following are equivalent for a group $$G$$:


 * 1) $$G$$ is a 3-abelian group, i.e., the cube map of $$G$$ is an endomorphism of $$G$$.
 * 2) $$G$$ is a 2-Engel group (also called a Levi group) and the derived subgroup of $$G$$ is either trivial or has exponent dividing three.

Applications

 * 3-abelian implies class three