TITLE: Free groups and free restricted Lie algebras
QUESTION [13 upvotes]: 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:

$\delta_1(G) = G$, and
$[\delta_k(G),\delta_{\ell}(G)] \subset \delta_{k+\ell}(G)$ for all $k,\ell \geq 1$, and
$(\delta_k(G))^p \subset \delta_{pk}(G)$ 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?

REPLY [6 votes]: 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.