Transitive factorizations of free partially commutative monoids and Lie algebras

Let $\M(A,\theta)$ be a free partially commutative monoid. We give here a necessary and sufficient condition on a subalphabet $B\subset A$ such that the right factor of a bisection $\M(A,\theta)=\M(B,\theta\_B).T$ be also partially commutative free. This extends strictly the (classical) elimination theory on partial commutations and allows to construct new factorizations of $\M(A,\theta)$ and associated bases of $L\_K(A,\theta)$.