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We study pre-Lie pairs, by which we mean a pair of a homotopy Lie algebra anda pre-Lie algebra with a compatible pre-Lie action. Such pairs provide a wealthof algebraic structure, which in particular can be used to analyze the homotopyLie part of the pair.  Our main application and the main motivation for this development are the dgLie algebras of hairy graphs computing the rational homotopy groups of themapping spaces of the $E_n$ operads. We show that twisting with certainMaurer-Cartan elements trivializes their Lie algebra structure. The result canbe used to understand the rational homotopy type of many connected componentsof the mapping spaces of $E_n$ operads.

Pre-Lie pairs and triviality of the Lie bracket on the twisted hairy graph complexes