I asked the following question in math.stackexchange and did not get any answer, so I am asking it here:

Let $G$ be a group and let $G_n$ be its series of dimension subgroups defined as follows:

$$ G _ n=\{ g \in G | g-1 \in \Delta^n \} $$ where $\Delta$ is the augmentation ideal.

This series has the property $[G_m,G_n]\leq G_{m+n}$. If $G_n/G_{n+1}$ are elementary abelian p groups It follows that $$ L(G)= \bigoplus (G_n/G_{n+1}) $$ forms a Lie algebra over $F_p$.

In a paper I am reading it is said that $L(G)$ and $L(\hat G)$ are isomorphic. (The group in consideration is a 2 group). Does this follow easily from the definitions? If not could you give me a reference for this?