Let $\mathfrak g$ be a finite-dimensional real [Lie algebra](/page/Lie%20Algebra), and let $\widetilde G$ be a connected and simply connected Lie group with identity element $\widetilde e$ whose Lie algebra $T_{\widetilde e}\widetilde G$ is identified with $\mathfrak g$. Then every connected Lie group $G$ with identity element $e_G$ whose Lie algebra $T_{e_G}G$ is identified with $\mathfrak g$ is isomorphic, as a Lie group inducing the prescribed Lie algebra identification, to a quotient

\begin{align*} \widetilde G/\Gamma \end{align*}

for some discrete subgroup $\Gamma \le Z(\widetilde G)$.

Conversely, for every discrete subgroup $\Gamma \le Z(\widetilde G)$, the quotient $\widetilde G/\Gamma$ is a connected Lie group, the quotient homomorphism $q_\Gamma:\widetilde G\to \widetilde G/\Gamma$ is a covering homomorphism, and the differential $d(q_\Gamma)_{\widetilde e}$ identifies the Lie algebra of $\widetilde G/\Gamma$ with $\mathfrak g$. Moreover, if $\Gamma_1,\Gamma_2 \le Z(\widetilde G)$ are discrete subgroups, then $\widetilde G/\Gamma_1$ and $\widetilde G/\Gamma_2$ are isomorphic by a Lie group isomorphism whose differential at the identity induces the identity map on $\mathfrak g$ if and only if $\Gamma_1=\Gamma_2$.