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 $$m$$:

$$\gamma_m(L(G))/\gamma_{m+1}(L(G)) \cong \gamma_m(G)/\gamma_{m+1}(G)$$