Transitive factorizations of free partially commutative monoids and Lie algebras

Let M(A, θ) be a free partially commutative monoid. We give here a necessary and sufficient condition on a subalphabet B ⊂ A such that the right factor of a bisection M(A, θ) = M(B, θB).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, θ) and associated bases of LK(A, θ).