Explicit description of lower central series of associated Lie ring of a group

Statement

Suppose $G$ is a group and $L(G)$ is its associated Lie ring. Recall that, as an additive group:

$L(G)=\bigoplus _{i=1}^{\infty }\gamma _{i}(G)/\gamma _{i+1}(G)$

where $\gamma _{i}(G)$ are the members of the lower central series of $G$, starting at $\gamma _{1}(G)=G$, $\gamma _{2}(G)=[G,G]$, and so on. Each of the quotients is an abelian group, hence can be inserted into the above summation. Note that the Lie ring structure is not as a direct sum.

Then, for any positive integer $m$, the $m^{th}$ member of the lower central series of $L(G)$ equals the part of the summation that starts from $m$ onward. Explicitly, it is the subgroup:

$\gamma _{m}(L(G))=\bigoplus _{i=m}^{\infty }\gamma _{i}(G)/\gamma _{i+1}(G)$

Note that the Lie ring structure is not as a direct sum, but rather, obtained by restricting the Lie ring structure from $L(G)$.

Further, we have that, for any nonnegative integer $m$:

$\gamma _{m}(L(G))/\gamma _{m+1}(L(G))\cong \gamma _{m}(G)/\gamma _{m+1}(G)$