Let $G$ be a connected Lie group with identity element $e$, and let $p:\widetilde G\to G$ be the universal covering map of the underlying smooth manifold of $G$, where $\widetilde G$ is equipped with its smooth covering-manifold structure. For every point $\widetilde e\in p^{-1}(e)$, there exists a unique Lie group structure on $\widetilde G$ such that $\widetilde e$ is the identity element and $p:\widetilde G\to G$ is a Lie [group homomorphism](/page/Group%20Homomorphism). With this Lie group structure, $\widetilde G$ is simply connected, and the differential

\begin{align*} d p_{\widetilde e}:T_{\widetilde e}\widetilde G\to T_eG \end{align*}

is an isomorphism of Lie algebras.