Exponent three implies class three for groups

Statement
Suppose $$G$$ is a group whose exponent is three. Then, $$G$$ is a group of nilpotency class three: it is a nilpotent group and its nilpotency class is at most three.

Related facts

 * Exponent two implies abelian, which follows from square map is endomorphism iff abelian

Facts used

 * 1) uses::Exponent three implies 2-Engel for groups (note that the analogous statement is not true for Lie rings)
 * 2) uses::2-Engel implies class three for groups (note that the analogous statement is true for Lie rings)

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