Let $G$ and $H$ two connected linear Lie groups. Is $G\ltimes H$ also linear?



Semidirect product of two linear groups
No. There is a semidirect product $\mathbf{R}\ltimes (\mathbf{R}\times\mathbf{R}/\mathbf{Z})$ that is not linear. Namely, the quotient of the Heisenberg group by a lattice of its center has this form.


Replacing $\mathbf{R}$ with $\mathbf{C}$ also yields an example in the realm of complex Lie groups.


These are classical examples of non-linear Lie groups. Indeed, if $G$ is a solvable connected Lie group then (the closure of) its derived subgroup is simply connected (indeed, taking a complex representation, one can triangulate it, and this directly follows since every closed connected subgroup of the upper unipotent group is simply connected).


