Consider the Baker-Campbell-Hausdorff formula $\Phi(X,Y)\in\mathbb{Q}\ll X,Y\gg$ in non-commutative variables. Define $X*Y:=\Phi(X,Y)$ and $[X,Y]=(-X)*(-Y)*X*Y$, and then as usual define for any vector
$\mathbf{e}=(e_1,\ldots,e_r)\in\mathbb{N}^r$ the repeated commutator

$$[X,Y]{\mathbf{e}}:=[X,\underbrace{Y,\ldots,Y}_{e_1},\underbrace{X,\ldots,X}_{e_2},\ldots]$$ (here $[X_1,\ldots,X_r]$ is defined as $[[X_1,\ldots,X_{r-1}],X_r]$).

I think that there is a an analogous of the BCH formula on expressing $XY-YX$ in terms on the commutators $[X,Y]_\mathbf{e}$. That is, if for $\mathbf{e}=(e_1,\ldots,e_r)$ we define $<\mathbf{e}>=e_1+\ldots+e_r$ then there exist rational numbers $t_\mathbf{e}$ for all $\mathbf{e}\in\mathbb{N}^r$ and for all $r$ such that if we put $v_n(X,Y)=\sum_{<\mathbf{e}>=n}[X,Y]_\mathbf{e}$ then

$$XY-YX=\sum_{n\in\mathbb{N}}v_n(X,Y)$$.

I would appreciate any reference about this.