Non-trivial open connected subgroup of a connected Lie Group 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!

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 $H$.
