Non-trivial open connected subgroup of a connected Lie Group

Question

Let $G$ be a connected Lie group. Then, I would like to show that there does not exist a open connected Lie subgroup $H$ such that $\{e\}\subsetneq H \subsetneq G$.

Any help or hint would be very helpful!

Answer 1

That's true because, on any connected Lie group, any neighborhood of $e$ spans the whole group. So, if $H$ is open, it is a neighborhhod of $e$ and therefore the subgroup spanned by $H$ (which is $H$ itself) is the whole group $G$.

Answer 2

It is not hard to show that open subgroups are also topologically closed. Now your intermediate subgroup and its complement separate $G$, contradicting connectivity. You do not need the connectedness assumption on $H$.