Let $X$ be an alphabet, ${\sf Lyn}$ be the set of Lyndon words on $X$ and $L$ be the free Lie ring on $X.$ For $w\in {\sf Lyn}$ we denote by $[w]$ the corresponding element of the Lyndon-Shirshov basis. Then $\{[w]\mid w \in {\sf Lyn}\}$ is a basis of $L$ considered as a free abelian group. Therefore, for any $u,v\in {\sf Lyn}$ there exist a unique family of integers $\{c^w_{u,v}\}_{w\in {\sf Lyn}}$ such that $$[[u],[v]]=\sum_{w\in {\sf Lyn}} c_{u,v}^w [w].$$
What is known about $c^w_{u,v}$?
Is there a way to compute them without huge straightforward computations?
