Here is a simple question regarding the standard Lyndon basis for the free Lie Algebra. Suppose I take two lyndon words $m$ and $n$ and their standard bracketings $B(m)$ and $B(n)$ as elements in the free lie algerba. Suppose further that $m < n$, so that $mn$ is a Lyndon word.

My question is when we express the bracket $[B(m),B(n)]$ in the Lyndon basis, is it of the form

$[B(m),B(n)] = B(mn) + \sum_{l>mn, |l| = |m|+|n|} a_l B(l)$