TITLE: Criterion to construct a $\mathbb{Z}$-basis of a free $\mathbb{Z}$-Lie algebra
QUESTION [5 upvotes]: Let $L(\mathbb{Z},n)$ (resp. $L(\mathbb{Q},n)$) be the free Lie algebra over $\mathbb{Z}$ (resp. over $\mathbb{Q}$) with generating set $\{x_1,\dots,x_n\}$. 
Let $\mathcal B$ be a $\mathbb{Q}$-basis of $L(\mathbb{Q},n)$ consisting of elements of the form $[x_{i_1},[x_{i_2},\dots,[x_{i_{r-1}},x_{i_r}]\dots]]$ for some $i_1,\dots,i_r\in\{1,\dots,n\}$. Is $\mathcal B$ automatically a $\mathbb{Z}$-basis of $L(\mathbb{Z},n)$?

REPLY [2 votes]: It turns out my intuition was wrong and the answer is "no".
Take for instance $n=3$, and attribute the degree $\alpha_i$ to $x_i$, so that $L(\mathbb{Q},3)$ becomes graded by the free $\mathbb{Z}$-module $\oplus_{i=1}^3\mathbb{Z}\alpha_i$. Then the homogeneous component of degree $2\alpha_1+\alpha_2+\alpha_3$ of $L(\mathbb{Q},3)$ has dimension $3$, and admits the $\mathbb{Q}$-basis $\mathcal B=\{[x_1,[x_1,[x_2,x_3]]], [x_2,[x_1,[x_1,x_3]]],[x_3,[x_1,[x_1,x_2]]]\}$. However,
$$[x_1,[x_3,[x_1,x_2]]]=-\frac{1}{2}[x_1,[x_1,[x_2,x_3]]]+\frac{1}{2}[x_2,[x_1,[x_1,x_3]]]+\frac{1}{2}[x_3,[x_1,[x_1,x_2]]].$$