I am trying read the proof of the following theorem.
Let $G$ be a Lie group with Lie algebra $\mathfrak g$ and $\mathfrak h$ be a Lie subalgebra of $\mathfrak g.$ Then there exists a unique connected Lie subgroup $H$ of $G$ with Lie algebra $\mathfrak h.$
The proof is as follows first define an involutive smooth distribution $a\mapsto D_a$ on $G$ as $D_a:=DL_a(1)\mathfrak h.$ Then following any standard textbook, one can use Forbenius theorem to obtain a unique connected maximal immersed submanifold $H$ such that $H$ is an integral submanifold of $D$ at $1\in G.$ Again, by using translation and uniqueness one can sow that $H$ is indeed a Lie subgroup of $G.$ Now comes the uniqueness part where I am stuck! Let $H^\prime$ be another cnnected Lie subgroup of $G$ such that $H^\prime$ has Lie algebra $\mathfrak h.$ Once again one can show that connect immersed submanifold $H^\prime$ is an integral submanifold of $D$ at $1\in G.$ Therefore, we have $H^\prime\subseteq H.$ Also, can show that the inclusion map $i:H^\prime\to H$ is indeed smooth and local diffeomorphism at each point of $H^\prime.$ Now I'll be done if can obtain a set $U$ containing $1$ such that $U$ is open in both $G,$ $H$ and $H^\prime$ and $U\subseteq H^\prime.$ But I do not know how to obtain that!! I can obtain that $H^\prime$ is open in $H$ since $i$ is a submersion at every point. But now if I want to use the fact that for any connected topological group the open neighourhoods of identity generates the group, I have a problem, which is I know $H$ is connected in the subspace topology of $G$ but the topology of $H$ may be finer than that!!
P.S. This is not a homework problem. The problem is he uniqueness part is left for the readers in the book I a reading which is the book of Kumaresan.