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