Koszul duality and the Poincar\'e-Birkhoff-Witt theorem

Using a homotopy introduced by de Wilde and Lecomte and homological perturbation theory for A_-algebras, we give an explicit proof that the universal enveloping algebra UL of a differential graded Lie algebra L is Koszul, via an explicit contracting homotopy from the cobar construction CL of the Chevalley-Eilenberg chain coalgebra CL of L to UL.