Let $k$ be a field of characteristic 0. Let $\mathtt{DGA}_{k}$ denote the category of DG algebras and $\mathtt{DGLA}_{k}$ denote the category of DG Lie algebras. It is well known that there are model structures on them that the weak equivalences are the quasi-isomorphisms and the fibrations are the maps which are degreewise surjective.
We have a pair of adjoint functors (in fact a Quillen pair) $$\mathcal{U}: \mathtt{DGLA}_{k} \leftrightarrows\mathtt{DGA}_{k} : \mathcal{Lie}$$ Where $\mathcal{U}$ is the universal enveloping algebra functor and $\mathcal{Lie}$ is the forgetful functor that sends every DG algebra to the DG Lie algebra defined by commutator bracket.
I am thinking about some derived functors on those model categories. One thing I need to convince myself is that $\mathcal{U}$ preserves quasi-isomorphism. The only proof I can find is in Félix, Halperin and Thomas' book Rational Homotopy Theory, but it is kind of brief and I can't fully understand the argument.
Because we are working over a field of characteristic 0, the homology $H$ will be a direct summand of our DG Lie algebra $L$ as complexes. And by PBW theorem, we can identify $\mathcal{U}(L)$ with $\mathrm{Sym}(L)$ as complexes. But now we need to show that $\mathrm{Sym}$ preserves quasi-isomorphism, and I don't know how to show it. Can anyone give me some hints or some good references? Thank you very much.