The Milnor-Moore theorem for $L_{\mathrm{\infty }}$ algebras in rational homotopy theory

We give a construction of the universal enveloping $A_{\mathrm{\infty }}$ algebra of a given $L_{\mathrm{\infty }}$ algebra, alternative to the already existing versions. As applications, we derive a higher homotopy algebras version of the classical Milnor-Moore theorem, proposing a new $A_{\mathrm{\infty }}$ model for simply connected rational homotopy types, and uncovering a relationship between the higher order rational Whitehead products in homotopy groups and the Pontryagin-Massey products in the rational loop space homology algebra.