About connected Lie Groups
An open subgroup $H$ of a topological group $G$ is closed because $$ G \smallsetminus H = \bigcup_{g \notin H} gH $$ is open as union of the open sets $gH$.
Now take your neighborhood $U$ of the identity, let $H = \bigcup_{n \in \mathbb{Z}} U^{n}$ and check that $H$ is an open (hence closed) subgroup of $G$. By connectedness $G = H$.