Over a field of characteristic zero we prove two formality conditions. We prove that a dg Lie algebra is formal if and only if its universal enveloping algebra is formal. We also prove that a commutative dg algebra is formal as a dg associative algebra if and only if it is formal as a commutative dg algebra. We present some consequences of these theorems in rational homotopy theory.

We construct a dg" role="presentation" style="display: inline; line-height: normal; font-size: 17.3333px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(64, 64, 64); font-family: "Times New Roman", Times, serif; position: relative;">dgdg Lie algebra model for the universal cover of the classifying space of the grouplike monoid of homotopy automorphisms of a space that fix a given subspace. We derive the model from a known model for based homotopy automorphisms together with general result on rational models for geometric bar constructions.