Filtered curved Lie algebras are quite common in mathematics. A typical example is graded Lie algebra of differential forms on a complex manifolds with values in the endomorphism bundle of a holomorphic vector bundle equipped with a connection compatible with the complex structure; the filtration is the natural generalization of Hodge filtration on scalar valued differential forms. Filtered here means equipped with a decreasing filtration \(F_pL\) with \(p\) non negative integer. In many geometric examples, the first quotient \(F_0L/F_1L\) is a DG-Lie algebra that controls an interesting deformation problem. The aim of the talk is to illustrate a general algebraic procedure for the construction of morphisms annihilating obstructions to deformations. This is done by extending in a relative case a previous result by Bandiera, Lepri and myself (2021) and then applying the PBW isomorphism in the (coalgebra) version described by Quillen (Rational homotopy theory).