A doubt on Lie subgroups of Lie groups The following theorem can be found in Lee's ``Introduction to Smooth Manifold".
Suppose $G$ is a Lie group with Lie algebra $\mathfrak g.$ If $\mathfrak h$ is a Lie subalgebra of $\mathfrak g$ then there is a unique connected Lie subgroup $H$ of $G$ with Lie algebra $\mathfrak h.$
Are the following statements true? Given the hypothesis of the above theorem, there is a subgroup $H$ of $G$ such that $H$ can be given a topology and a smooth structure such that the natural inclusion map $i:H\to G$ is a Lie group homomorphism and $Di(e)T_{e}H=\mathfrak h.$
My question is in which topology $H$ is connected? Does uniqueness mean that if there is another subgroup $H^\prime$ with a topology and smooth structure, connected in its own topology and $Di(e)T_{e}H=\mathfrak h$ then $H=H^\prime$ and the identity map from $H$ to $H^\prime$ is a Lie group isomorphism?
Can there be some natural counterexamples violating the asumptions?
Definition: Let $M$ be a smooth manifold. Then $S\subseteq M$ is said to be a submanifold if $S$ has its own topology and smooth structure with respect to the inclusion map is a smooth immersion.
Definition: Let $G$ be a Lie group. A subgroup $H\subseteq G$ is called a Lie subgroup if $H$ is a submanifold and Lie group with respect to the smooth structure of being a submanifold.
 A: I don't have a copy of the book accessible to me at the moment. I think it is a good question. Let me explain a standard "pathological" example first.
Let $V = \mathbb{R}^2$ and $\mathbb{Z}^2 \subset \mathbb{R}^2$ be the integer lattice inside $V$. Then $T^2 := V/\mathbb{Z}^2 \simeq U(1) \times U(1)$ is a (abelian) two-dimensional Lie group, where the Lie group "product" is induced by addition of vectors in $V$.
At $\mathbf{0} \in V$, consider a tangent line $l \subset T_\mathbf{0}(\mathbb{R}^2) = \mathbb{R}^2$. If the slope of $l$ is rational, then $l \mod \mathbb{Z}^2$ defines a Lie subgroup of $T^2$ with the subspace topology (it is a closed submanifold of $T^2$). But if the slope of $l$ is irrational, then $l \mod \mathbb{Z}^2$ defines a Lie subgroup of $T^2$ which is not a closed submanifold.
It is due to pathological examples such as this one that many authors define the notion of a Lie subgroup without requiring that a Lie subgroup have the induced topology.
I don't know the precise definitions that Lee is using in his book, so I cannot discuss further before knowing the precise definitions he uses for a "Lie subgroup" and for a "submanifold".
If the OP wishes to discuss further, please add these definitions to your post, and then I may be able to discuss in more detail.
