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

Statement
Suppose $$L$$ is a (1,1)-bi-Engel Lie ring, i.e., $$u,x],[u,y = 0$$ for all $$u,x,y \in L$$. Then, we have that $$2L,L],[L,L = 0$$. More explicitly:

$$2a,b],[c,d = 0 \ \forall \ a,b,c,d \in L$$

Proof
Given: Lie ring $$L$$. $$u,x],[u,y = 0$$ for all $$u,x,y \in L$$.

To prove: $$2a,b],[c,d = 0 \ \forall \ a,b,c,d \in L$$

Proof: Fix $$a,b,c,d \in L$$ for this proof.