N-nilpotent group

Definition
Suppose $$G$$ is a group and $$n$$ and $$c$$ are integers, with $$c$$ nonnegative. We say that $$G$$ is $$n$$-nilpotent with class at most $$c$$ if the $$(c+1)^{th}$$ member of the n-lower central series of $$G$$ is the trivial subgroup. The n-lower central series is a series starting with the first member $$G$$, where each member is defined as the subgroup generated by all n-commutators between elements of the previous member and all of $$G$$. Here, the $$n$$-commutator of $$x,y \in G$$ is defined as $$[x,y]_n := (xy)^ny^{-n}x^{-n}$$.

Terminology caution
Note that this differs from the notion of p-nilpotent group. Thus, when somebody uses $$n$$-nilpotent for a specific numerical value of $$n$$ that is prime, it is more likely that they are referring to the concept of p-nilpotent than the concept of n-nilpotent mentioned here.