Global Lazard Lie group

Quick definition
A group is termed a global Lazard Lie group if its defining ingredient::nilpotency class is finite and less than or equal to the group's defining ingredient::powering threshold.

Explicit definition
A group $$G$$ is termed a global class $$c$$ Lazard Lie group for some natural number $$c$$ if 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.

A group is termed a global Lazard Lie group if it is a global class $$c$$ Lazard Lie group for some natural number $$c$$.

A global Lazard Lie group is a group that can participate on the group side of the global Lazard correspondence. The Lie ring on the other side is its global Lazard Lie ring.

Set of possible values $$c$$ for which a group is a global class $$c$$ Lazard Lie group
A group is a global Lazard Lie group if and only if its nilpotency class is less than or equal to its powering threshold. The set of permissible $$c$$ values for which the group is a global class $$c$$ Lazard Lie group is the set of $$c$$ satisfying:

nilpotency class $$\le c \le$$ powering threshold

p-group version
A p-group is termed a global Lazard Lie group if its defining ingredient::nilpotency class is at most $$p - 1$$.