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