2-Engel and 3-torsion-free implies class two for Lie rings

Statement

Suppose $L$ is a Lie ring that satisfies the following two conditions:

• $L$ is a 2-Engel Lie ring, i.e., $[x,[x,y]]=0$ for all $x,y\in L$.
• $3x=0$ implies $x=0$, i.e., $L$ is free of 3-torsion.

Then, $L$ is a Lie ring of nilpotency class two, i.e., $[x,[y,z]]=0$ for all $x,y,z\in L$.

Related facts

Similar facts for 2-Engel conditions

• 2-Engel and 3-torsion-free implies class two for groups
• 2-Engel and Lazard Lie ring implies class two
• 2-Engel and Lazard Lie group implies class two
• 2-Engel implies class three for Lie rings
• 2-Engel implies class three for groups
• Nilpotency class three is 3-local for Lie rings

Similar facts for higher Engel conditions

• 3-Engel and 2-torsion-free implies 2-local class three for Lie rings
• 3-Engel and (2,5)-torsion-free implies class six for Lie rings

Facts used

1. 2-Engel implies third member of lower central series is in 3-torsion for Lie rings

Proof

Given: A 2-Engel Lie ring $L$ that is 3-torsion-free.

To prove: $[x,[y,z]]=0$ for all $x,y,z\in L$.

Proof: By Fact (1), we have that $3[x,[y,z]]=0$ for all $x,y,z\in L$. The result now follows immediately from the 3-torsion-free assumption.