Finite covering of linear simisimple Lie group

Question

Let $G$ be a semisimple real Lie group such that there exits a linear semisimple Lie group $\tilde G$ that finitely covers $G$. Is $G$ also linear?

Answer 1

I am not sure if there is a simpler proof in the case of finite normal subgroups, but here is a general theorem, see Theorem 11.5 in

J.Humphreys, "Linear Algebraic Groups."

Theorem. Suppose that $G$ is an algebraic linear Lie group, $H\triangleleft G$ is a closed normal subgroup. Then $G/H$ is again linear.

In your case, you can replace your $G$ with its (Zariski) closure and use the fact that finite groups are closed.