Let $R$ be a graded commutative unital ring and $M$ a graded $R$-module. 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.

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.