This is a question with a proof needed in case of a positive answer.
Let $G$ be a Lie group and $H$ a path-connected normal subgroup of $G$, so that the identity in $G$ is also in $H$. Is $H$ the connected component of identity in $G$? If so, what would be a proof? If not, what other conditions on G, H or their factor group are needed to make the answer into a “yes”?
No. Let $G=GL(2,\mathbb{R})$ and let $H=\left\{\left(\begin{smallmatrix}1&0\\0&1\end{smallmatrix}\right)\right\}$. Then $H$ is a path-connected normal subgroup of $G$, but the connected component of identity in $G$ is $G$ itself.