First of all, sorry for using this account. Somehow I can't login to my previous one anymore and am thus using the account associated to my MSE one. Also, I already asked the question on MSE, but didn't get any answer or comment yet, maybe due to the vague formulation.

In the "Lectures on the Fourteenth Problem of Hilbert" by Nagata, he mentioned that if $G$ is a complex connected semisimple Lie group, then the answer to the following problem is "yes". This is supposed to be proven in Weyl's "Classical Groups".

Let $G\subseteq\operatorname{GL}_n(\mathbb{C})$ act on $\mathbb{C}[x_1,\dots,x_n]$ via $x\mapsto A\cdot x$ for $A\in G$. Is the invariant ring $\mathbb{C}[x_1,\dots,x_n]^G$ a finitely generated $\mathbb{C}$-algebra?

I have no familiarity at all with Lie groups, and just wish to reference the result for the sake of completeness. But I can't seem to find it in the above book. Does anyone of you maybe know which result exactly is meant? Also, is there a nice classification of the subgroups of $\operatorname{GL}_n$ which are connected semisimple Lie groups? Or rather, what are "known" examples of such groups?

Thank you very much in advance!