Connected Lie groups are central quotients of simply connected integrations

Let $\mathrm{\Gamma }=\ker p$. A fiber of a covering is discrete, so $\mathrm{\Gamma }$ is discrete; it is normal because it is a kernel. For fixed $\gamma \in \mathrm{\Gamma }$, the map $x\mapsto x\gamma x^{-1}$ is continuous from connected $\tilde{G}$ into the discrete space $\mathrm{\Gamma }$, hence constant. At the identity its value is $\gamma$, so $\gamma$ is central.

The fibers of $p$ are exactly the cosets of $\mathrm{\Gamma }$. Hence $p$ factors through a bijective homomorphism $\bar{p}:\tilde{G}\mathrm{/}\mathrm{\Gamma }\to G$. Covering charts for $p$ give the quotient its unique smooth structure for which the quotient projection is a local diffeomorphism, and in those charts $\bar{p}$ and its inverse are smooth. Thus $\bar{p}$ is a Lie-group isomorphism.

Conversely, let $\mathrm{\Gamma }$ be a discrete central subgroup of a simply connected Lie group $\tilde{G}$. It is closed and embedded by [L3]. Choose an identity neighborhood meeting $\mathrm{\Gamma }$ only in the identity and shrink it so that distinct translates are disjoint. Its translates furnish smooth quotient charts, making $\tilde{G}\to \tilde{G}\mathrm{/}\mathrm{\Gamma }$ a covering homomorphism. Its identity differential is an isomorphism, so the two groups have the same Lie algebra. The trivial subgroup and one-point group are included. Countable choice is used exactly through [L3]; steps 1.1–2.1 need no additional choice.