(1,1)-bi-Engel and 2-torsion-free implies metabelian

Statement
Suppose $$L$$ is a Lie ring satisfying the following two conditions:


 * 1) $$L$$ is a uses property satisfaction of::(1,1)-bi-Engel Lie ring: $$u,x],[u,y = 0$$ for all $$u,x,y \in L$$.
 * 2) $$L$$ is 2-torsion-free: $$2a = 0 \implies a = 0$$ for $$a \in L$$.

Then, $$L$$ is a proves property satisfaction of::metabelian Lie ring, i.e., the second derived subring $$L,L],[L,L$$ is zero.

Facts used

 * 1) uses::(1,1)-bi-Engel implies second derived subring is in 2-torsion

Proof
The proof follows directly from Fact (1).