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Over a field of characteristic zero we prove two formality conditions. Weprove that a dg Lie algebra is formal if and only if its universal envelopingalgebra is formal. We also prove that a commutative dg algebra is formal as adg 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.

Non-commutative formality implies commutative and Lie formality