Bi-Engel Lie ring

For a pair of values
Suppose $$i,j$$ are nonnegative integers. An $$(i,j)$$-bi-Engel Lie ring is a Lie ring $$L$$ with the property that for any elements $$u,x,y \in L$$, we have:

$$[(\operatorname{ad}u)^ix,(\operatorname{ad}u)^jy] = 0$$

where $$\operatorname{ad}$$ denotes the adjoint action.

Note that being an $$(i,j)$$-bi-Engel Lie ring is equivalent to being a $$(j,i)$$-bi-Engel Lie ring.

For a single number
Suppose $$n$$ is a nonnegative integer. Then:


 * A $$n$$-bi-Engel Lie ring is a Lie ring $$L$$ that is an $$(i,j)$$-bi-Engel Lie ring for all pairs of nonnegative integers $$i,j$$ with $$i + j \ge n$$.
 * A weak $$n$$-bi-Engel Lie ring is a Lie ring $$L$$ that is an $$(i,j)$$-bi-Engel Lie ring for all pairs of positive integers $$i,j$$ with $$i + j \ge n$$.

A bounded bi-Engel Lie ring is a Lie ring that is $$n$$-bi-Engel for some natural number $$n$$. The smallest $$n$$ that works is the bi-Engel degree of the Lie ring.

For a pair of numbers
Because $$(i,j)$$-bi-Engel equals $$(j,i)$$-bi-Engel, we only list the $$i \ge j$$ cases.