2-Engel and 3-torsion-free implies class two for groups

Statement
Suppose $$G$$ is a group that satisfies the following two conditions:


 * $$G$$ is a uses property satisfaction of::Levi group (also called 2-Engl group), i.e., $$[x,[x,y]]$$ is the identity element for all $$x,y \in G$$.
 * $$G$$ has no non-identity element of order three.

Then, $$G$$ is a proves property satisfaction of::group of nilpotency class two, i.e., $$[x,[y,z]]$$ is the identity element for all $$x,y,z \in G$$.

Facts about 2-Engel

 * 2-Engel implies class three for groups, 2-Engel implies class three for Lie rings
 * 2-Engel and 3-torsion-free implies class two for Lie rings
 * 2-Engel and Lazard Lie ring implies class two
 * 2-Engel and Lazard Lie group implies class two

Facts about other Engel conditions

 * 3-Engel and (2,5)-torsion-free implies class four for groups
 * 4-Engel and (2,3,5)-torsion-free implies class seven for groups