M. Manetti: A relative version of the ordinary perturbation lemma. Rend. Mat. Appl. (7) 30 (2010) 221-238; pdf

Abstract. The perturbation lemma and the homotopy transfer for $L_{\infty}$-algebras is proved in a elementary way by using a relative version of the ordinary perturbation lemma for chain complexes and the coalgebra perturbation lemma.