Plotkin’s problems for general Lie rings

Determine, for arbitrary Lie rings, whether D_ω(L)=γ_ω(L) and whether, for every positive integer c, there exists an integer m=m(c) such that D_m(L)=0 for every nilpotent Lie ring of class at most c.

Background

The dimension series and lower central series of a Lie ring are defined by D_n(L)=L\cap\varpi(L)^n and γ{n+1}(L)=[γ_n(L),L]. The paper considers Lie-ring analogues of Plotkin’s two group problems: equality of the limiting intersections and uniform eventual vanishing in bounded nilpotency class. It explicitly states that the analogues remain open in general, although the paper resolves both for metabelian Lie rings by proving D{2c-1}(L)=0 for metabelian Lie rings of class at most c and then deducing D_ω(L)=γ_ω(L).