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

Related facts

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