If $G$ is any group and $\gamma\_k(G)$ denotes the $k$th term in the lower central series of $G$, then the commutator bracket on $G$ endows


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


with the structure of a Lie ring. A famous theorem that should probably be attributed to Magnus says that if $G$ is the free group on $n$ generators, then $\mathcal{L}(G)$ is the free Lie ring on $n$ generators. This result is documented in many places, e.g. Magnus--Karass--Solitar's book on combinatorial group theory and Serre's book Lie Group and Lie Algebras.


Now fix a prime $p$ and let $\delta\_k(G)$ be the fastest descending central series of $G$ satisfying the following three properties:


1. $\delta\_1(G) = G$, and
2. $[\delta\_k(G),\delta\_{\ell}(G)] \subset \delta\_{k+\ell}(G)$ for all $k,\ell \geq 1$, and
3. $(\delta\_k(G))^p \subset \delta\_{pk}(G)$ for all $k \geq 1$.


This series was first defined by Zassenhaus. The commutator bracket and the $p$th power operation on $G$ endow


$$\Lambda(G) = \bigoplus\_{k=1}^{\infty} \delta\_k(G) / \delta\_{k+1}(G)$$


with the structure of a restricted Lie algebra over the field $\mathbb{F}\_p$.


I'm pretty certain that I can prove that if $G$ is a free group on $n$ generators, then $\Lambda(G)$ is the free restricted Lie algebra on $n$ generators over the field $\mathbb{F}\_p$. I need this result for a paper I am writing (on an unrelated topic), but I'm certain that this result is known and would greatly prefer to just cite a reference for it.


**Question**: Can anyone give me a reference for this?



Free groups and free restricted Lie algebras
This is Theorem 6.5 p.130 from Lazard, Michel.
Sur les groupes nilpotent et les anneaux de Lie.
Ann. Sci. Ecole Norm. Sup (3) 71 (1954) 101-190.


