Engel Lie ring

Definition in terms of inner derivations
A Lie ring is termed a $$n$$-Engel Lie ring if every inner derivation of the Lie ring is a nilpotent derivation with nilpotency at most $$n$$.

Definition in terms of Lie brackets
A Lie ring $$L$$ is termed a $$n$$-Engel Lie ring if there exists a natural number $$n$$ such that for any $$x,y \in L$$, t:

$$[x,[x,\dots,[x,y]\dots]]$$

where $$x$$ occurs $$n$$ times.

Note that we sometimes simply use the term Engel Lie ring for a Lie ring where the value of $$n$$ may be dependent on $$x$$ and $$y$$. However, by Zelmanov's theorem on Engel Lie rings, this is equivalent to being a locally nilpotent Lie ring.

Facts

 * Engel's theorem: This states that if $$L$$ is a finite-dimensional Lie algebra over a field, the Engel property is equivalent to being nilpotent.
 * Zelmanov's theorem on Engel Lie rings: This states that Engel Lie rings are locally nilpotent.
 * Kostrikin's theorem: This states that if $$L$$ is a $$d$$-generator Lie algebra over a field of characteristic $$p$$ satisfying the $$n$$-Engel condition for $$n < p$$ (or $$n$$ arbitrary in the case $$p = 0$$) then $$L$$ is nilpotent with nilpotency class bounded by a function of $$d$$ and $$n$$.