Global Lazard Lie ring

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

Explicit definition
A Lie ring $$L$$ is termed a global class $$c$$ Lazard Lie ring for some natural number $$c$$ if both the following hold:

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

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

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

Set of possible values $$c$$ for which a Lie ring is a global class $$c$$ Lazard Lie ring
A Lie ring is a global Lazard Lie ring 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 Lie ring is a global class $$c$$ Lazard Lie ring is the set of $$c$$ satisfying:

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