Associated graded Lie ring of a Lie ring

Definition
Let $$S$$ be a Lie ring. The associated graded Lie ring of $$S$$, sometimes termed the associated graded Lie ring, denoted $$L(S)$$, is a Lie ring defined as follows:


 * As an additive group, it is a direct sum of the successive quotients in the lower central series of $$S$$. If $$\gamma_1(S) = S$$ and $$\gamma_{i+1}(S) = [S,\gamma_i(S)]$$, the associated Lie ring of $$S$$ is, as an additive group:

$$L(S) = \bigoplus_i \gamma_i(S)/\gamma_{i+1}(S)$$


 * The Lie bracket is defined component-wise as follows. The Lie bracket of $$a \in \gamma_i(G)$$ and $$b \in \gamma_j(G)$$ is the Lie bracket $$[a',b']$$ as an element of $$\gamma_{i+j}(G)$$ (modulo $$\gamma_{i+j+1}(G)$$, where $$a', b'$$ are representatives of $$a$$ and $$b$$.

Dependence on quotient by nilpotent residual
The associated graded Lie ring of a group is the same as the associated Lie ring of the quotient ring of the Lie ring by its nilpotent residual. Thus, the notion is generally studied only for residually nilpotent Lie rings. Typically, we first replace a Lie ring by its quotient by its nilpotent residual before we commence study of the associated graded Lie ring.