Consider the fiber bundle $X\longrightarrow S^1\times S^1 $ with fibers $\mathbb R^q$.
If $X$ is a $G$-homogeneous space, where $G$ is a simply-connected and connected Lie group i.e $G$ acts transitively on $X$. My questions:
- Can always $X$ be written as $G/H$ for $G$ solvable?
- Is there any semisimple Lie group who can act transitively on $X$?