Let $G$ be a Lie group. Is it necessarily the case that there is a matrix Lie group $H$ and a smooth homomorphism $p:G\to H$ which is also covering map? And if so, what can be said about the kernel of $p$?
This "fact" is mentioned in the question here, I was hoping for a reference or sketch of the proof.
I know that there are Lie groups which are not matrix Lie groups, but that a version of Ado's Theorem says every Lie algebra is isomorphic to the Lie algebra of a matrix Lie group, and thus every Lie group is locally isomorphic to a matrix Lie group. As a novice in this area, I don't know how you would get a covering map from that.