TITLE: Free graded Lie algebras
QUESTION [11 upvotes]: Let $R$ be a graded commutative unital ring and $M$ a graded $R$-module (all gradings are over $\mathbb{Z}$). I'm looking for a reference for the following statement:

If $M$ is $R$-free, then the free graded Lie algebra over $R$ generated by $M$ is also $R$-free.

For the purposes of this question, a graded Lie algebra over $R$ is a graded $R$-module $L$ with an $R$-linear map $[-, - ] :  L\otimes_RL\to L$
satisfying the following identities for all homogeneous elements $a,b,c\in L$:
(1) $[a,b]+(-1)^{|a||b|}[b,a]=0$
(2) $(-1)^{|a||c|}  [a,[b,c]]   +  (-1)^{|b||a|}[b,[c,a]]   +    (-1)^{|c| |b|}  [c,[a,b]]      =    0  $
(3) $[a,a]=0$ if $a$ is of even degree
(4) $[a,[a,a]]=0$.
I should also say that I'm aware of Reutenauer's book, and, as far as I can tell, it does not deal with the graded case.

REPLY [4 votes]: I will answer under the assumption that $2$ is invertible in $R$, the general case of which follows after extension of scalars from the case $R=\mathbb{Z}[1/2]$, which is a particular case of Proposition 8.5.1 in Neisendorfer’s “Algebraic Methods in Unstable Homotopy Theory.” The assumption that $2$ is invertible is necessary to ensure that Definition 8.1.1 coincides with the definition stated above (see the following remark on p. 262). I do not know if this assumption is necessary.