Inner-Lazard Lie group

Quick definition
A group is termed a Lazard Lie group if its defining ingredient::3-local nilpotency class is finite and less than or equal to (the group's defining ingredient::powering threshold + 1).

Explicit definition
A group $$G$$ is termed an inner-Lazard Lie group if there is a natural number $$c$$ such that both the following hold:

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 group but larger and smaller values may not.

p-group version
A p-group is termed an inner-Lazard Lie group if its 3-local nilpotency class is at most $$p$$. In other words, every subgroup of it generated by at most three elements has nilpotency class at most $$p$$ where $$p$$ is the prime associated with the group.