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:

No.	Shorthand for property	Explanation
1	The additive group is a powered group for the set of all primes strictly less than $c$.	For any prime number $p<c$, and any element $a\in L$, there is a unique element $b\in L$ such that $pb=a$.
2	The 3-local nilpotency class is at most $c$.	For any subset of $L$ of size at most three, the subring of $L$ generated by that subset is a nilpotent Lie ring of nilpotency class at most $c$.

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 $p$.