Global LUCS-Lazard Lie ring

Definition
A Lie ring is termed a global LUCS-Lazard Lie ring if there exists a positive integer $$c$$ such that:


 * 1) $$G$$ is a nilpotent Lie ring of nilpotency class at most $$c$$.
 * 2) The following equivalent formulations:
 * 3) * For every positive integer $$i$$, every element of the $$i^{th}$$ lower central series member $$\gamma_i(L)$$ has a unique $$p^{th}$$ root in $$L$$ for all $$p \le i$$.
 * 4) * For every positive integer $$i$$, every element of the $$i^{th}$$ lower central series member $$\gamma_i(L)$$ has a unique $$p^{th}$$ root in the upper central series member $$Z^{c+1-i}(L)$$ for all $$p \le i$$.
 * 5) * For every positive integer $$i$$, every element of the $$i^{th}$$ lower central series member $$\gamma_i(L)$$ has unique $$p^{th}$$ roots in $$G$$ for all $$p \le i$$, and this root is inside $$Z^{c+1-i}(L)$$.