Must a lie subgroup of a simply connected lie group be necessarily embedded?

Question

Let $G$ be a simply connected lie group, must all lie subgroups of $G $ be embedded submanifolds?

Answer 1

First of all, one needs to be clear about the notion of a Lie subgroup: The use of this notion in the literature is inconsistent. From your post, it is clear that what you mean is:

Question. Suppose that $G, H$ are Lie groups, such that $G$ is simply connected, $f: H\to G$ is an injective morphism of Lie groups (i.e. a smooth monomorphism). Is $f$ an embedding?

The answer is easily negative. There are simpler examples with $H$ disconnected but I suspect that connectivity of $H$ was an unstated assumption.

Consider for instance $G=SU(2)\times SU(2)$ (it is simply connected). The group $G$ contains an (embedded) subgroup $T$ isomorphic to $S^1\times S^1$. The latter contains a dense Lie subgroup $H$ isomorphic to ${\mathbb R}$ (a line with irrational slope). Thus, $G$ contains a non-embedded (connected) Lie subgroup $H$.