Let $G$ be a finitely generated torsion-free nilpotent group. The Malcev completion of $G$ is a nilpotent Lie group $N$ into which $G$ embeds as a lattice. One way to construct this is to take the completion $\widehat{\mathbb{R}[G]}$ of the real group ring with respect to the augmentation ideal. This is a Hopf algebra, and $N$ is the set of group-like elements in it.

The Lie algebra of $N$ is the set of primitive elements $P(\widehat{\mathbb{R}[G]})$. Let $R$ be the subalgebra of $\widehat{\mathbb{R}[G]}$ generated by $P(\widehat{\mathbb{R}[G]})$. Question: is $R$ the universal enveloping algebra of $P(\widehat{\mathbb{R}[G]})$?

If I understand the setup correctly, Quillen proved in his paper "On the associated graded of a group ring" that the associated graded of $R$ is the universal enveloping algebra of the associated graded of $N$, but I don't know if that is true before we take the associated graded.

Of course, the field $\mathbb{R}$ is only playing a minor role here, and the natural question is to replace it by an arbitrary field of characteristic $0$.