Inner-Lazard Lie ring

Definition
An inner-Lazard Lie ring is a Lie ring $$L$$ such that the 3-local nilpotency class of $$L$$ is finite and is at most one more than the powering threshold of $$L$$.

Another way of putting this is that there must exist a natural number $$c$$ with both the following two properties:

Condition (1) gets more demanding (i.e., stronger, so satisfied by fewer groups) as $$c$$ increases, while condition (2) gets less demanding (i.e., weaker, so satisfied by fewer groups) as we increase $$c$$. Thus, a particular value of c may work for a Lie ring but larger and smaller values may not.

p-Lie ring
An inner-Lazard $$p$$-Lie ring is a special case of the above, namely a Lie ring $$L$$ such that:


 * 1) There is a prime $$p$$ such that every element of $$L$$ has order a power of $$p$$.
 * 2) The Lie subring of $$L$$ generated by any three elements of $$L$$ is a nilpotent Lie ring of nilpotency class at most $$p$$.