Given a simply connected locally compact group $G$, is it true that $G$ admits enough finite dimensional representations (over any field and not necessarily continuous) to separate points in $G$, what about over $\mathbb{C}$ and we require the representations to be continuous?


Again, this question is a follow-up of [this one](https://mathoverflow.net/questions/410186/examples-of-locally-compact-groups-that-do-not-admit-enough-finite-dimensional-r), and it seems better to ask it separately here.



Does every locally compact, simply connected group admit enough finite dimensional representations?
The connected Lie groups whose points are separated by the finite-dimensional complex representations are exactly the linear Lie groups, for instance by Th. 5.3 in [Beltiţă and Neeb - Finite-dimensional Lie subalgebras of algebras with continuous inversion](http://dx.doi.org/10.4064/sm185-3-3).


