Let $G$ be a connected Lie group and $H$ a normal connected subgroup. I'm looking for examples where $G\to G/H$ has a section as topological spaces, but not as Lie groups.

My motivation here is to find a continuous analogue of extensions of discrete groups, where splittings always exist on the level of sets, but only exist as groups when the extension is a semidirect product.

I've looked at exact sequences of Lie groups, such as: $$1\to SU(n) \to U(n) \to U(1) \to 1$$

But in this case, the sequence is split so $U(n) \cong SU(n)\rtimes U(1)$. I've also looked at: $$1\to SU(2) \to SO(4) \to SO(3) \to 1$$

I'm told in the comments that there is no splitting here on the level of groups. Am I correct that there is a topological splitting?

(Side question: does a topological splitting imply that $G\to G/H$ is a trivial fibration, so that $G$ is homeomorphic to $H\times G/H$?)

Ideally, I'd like to see an example where $H$ is abelian. I'd also like to consider situations where $H$ isn't normal so $G/H$ is just a homogeneous space and not a group, but I'm not completely sure how to phrase the question in that case. Cohomological criteria are welcome and appreciated.

Let $G$ be a connected Lie group and $H$ a normal connected subgroup. I'm looking for examples where $G\to G/H$ has a section as topological spaces, but not as Lie groups.

My motivation here is to find a continuous analogue of extensions of discrete groups, where splittings always exist on the level of sets, but only exist as groups when the extension is a semidirect product.

I've looked at exact sequences of Lie groups, such as: $$1\to SU(n) \to U(n) \to U(1) \to 1$$

But in this case, the sequence is split so $U(n) \cong SU(n)\rtimes U(1)$. I've also looked at: $$1\to SU(2) \to SO(4) \to SO(3) \to 1$$

But I believe there is still a splitting here as well. (I couldn't find a reference but this seems to be the case)

(Side question: does a topological splitting imply that $G\to G/H$ is a trivial fibration, so that $G$ is homeomorphic to $H\times G/H$?)

Ideally, I'd like to see an example where $H$ is abelian. I'd also like to consider situations where $H$ isn't normal so $G/H$ is just a homogeneous space and not a group, but I'm not completely sure how to phrase the question in that case. Cohomological criteria are welcome and appreciated.

Let $G$ be a connected Lie group and $H$ a normal connected subgroup. Are there examples where $G\to G/H$ has a section as topological spaces, but not as Lie groups?

My motivation here is to find a continuous analogue of extensions of discrete groups, where splittings always exist on the level of sets, but only exist as groups when the extension is a semidirect product.

I've looked at exact sequences of Lie groups, such as: $$1\to SU(n) \to U(n) \to U(1) \to 1$$

But in this case, the sequence is split so $U(n) \cong SU(n)\rtimes U(1)$. I've also looked at: $$1\to SU(2) \to SO(4) \to SO(3) \to 1$$

But here I believe this sequence is also split. (I can't find a reference for this so I could be mistaken)

(Side question: does a topological splitting imply that $G\to G/H$ is a trivial fibration, so that $G$ is homeomorphic to $H\times G/H$?)

Ideally, I'd like to see an example where $H$ is abelian. I'd also like to consider situations where $H$ isn't normal so $G/H$ is just a homogeneous space and not a group, but I'm not completely sure how to phrase the question in that case. Cohomological criteria are welcome and appreciated.

Let $G$ be a connected Lie group and $H$ a normal connected subgroup. I'm looking for examples where $G\to G/H$ has a section as topological spaces, but not as Lie groups.

My motivation here is to find a continuous analogue of extensions of discrete groups, where splittings always exist on the level of sets, but only exist as groups when the extension is a semidirect product.

I've looked at exact sequences of Lie groups, such as: $$1\to SU(n) \to U(n) \to U(1) \to 1$$

But in this case, the sequence is split so $U(n) \cong SU(n)\rtimes U(1)$. I've also looked at: $$1\to SU(2) \to SO(4) \to SO(3) \to 1$$

I'm told in the comments that there is no splitting here on the level of groups. Am I correct that there is a topological splitting?

(Side question: does a topological splitting imply that $G\to G/H$ is a trivial fibration, so that $G$ is homeomorphic to $H\times G/H$?)

Ideally, I'd like to see an example where $H$ is abelian. I'd also like to consider situations where $H$ isn't normal so $G/H$ is just a homogeneous space and not a group, but I'm not completely sure how to phrase the question in that case. Cohomological criteria are welcome and appreciated.