Left 2-Engel element

Definition
Suppose $$G$$ is a group and $$x$$ is an element of $$G$$. We say that $$x$$ is a left 2-Engel element if it satisfies the following equivalent conditions:


 * $$[x,[x,y]]$$ is the identity element for all $$y \in G$$.
 * $$[[y,x],x]$$ is the identity element for all $$y \in G$$.
 * The normal subgroup generated by $$x$$ is an abelian group.
 * $$x$$ commutes with all element in its conjugacy class in $$G$$.