2-Engel Lie ring implies third member of lower central series is in 3-torsion

Statement

Suppose $L$ is a Lie ring that is a 2-Engel Lie ring, i.e., $[x,[x,y]]=0$ for all $x,y\in L$. Note that, by equivalence of definitions of 2-Engel Lie ring, this is equivalent to saying that $[x,[y,z]]=[y,[z,x]]=[z,[x,y]]$ for all $x,y,z\in L$.

Then, the third member of the lower central series of $L$, i.e., $[L,[L,L]]$, is in the 3-torsion of $L$. In other words, $3[L,[L,L]]=0$.

Related facts

Applications

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

Proof

Given: A Lie ring $L$ such that $[x,[y,z]]=[y,[z,x]]=[z,[x,y]]$ for all $x,y,z\in L$.

To prove: $3[x,[y,z]]=0$ for all $x,y,z\in L$.

Proof: The proof follows by plugging the given data into Jacobi's identity, which states that:

$[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$ for all $x,y,z\in L$