Let $G$ be a Lie group with identity element $e$, and let $G_0$ denote the connected component of $G$ containing $e$. Then $G_0$ is a closed [normal subgroup](/page/Normal%20Subgroup) of $G$. Moreover, $G_0$ is open in $G$, and the connected components of $G$ are precisely the left cosets $gG_0$ for $g \in G$. Since $G_0$ is normal, these are also the right cosets $G_0g$.