This question already has an answer here:

In Nagata's "Lectures on the 14th problem of Hilbert" I found a reference to Weyl's "Classical Groups". Nagata writes that Weyl gives a positive answer to the original problem

If $G\subseteq\operatorname{GL}_n(\mathbb{C})$ is acting on $\mathbb{C}[x_1,\dots,x_n]$ via $x\mapsto A\cdot x$, is $\mathbb{C}[x_1,\dots,x_n]^G$ a finitely generated $k$-algebra?

in the special case that $G$ is a semisimple connected Lie group.

Now I have no familiarity whatsoever with Lie groups, and I would just like to reference this result. But I can't seem to find it in Weyl's book in this form. I guess it is formulated in a different way and the above is a more modern formulation. It would be nice if anyone familiar with Weyl's book or Lie groups could help me with this and maybe tell me which theorem this is. Also, what "known" groups fall under the notion "connected semisimple Lie group"?

Thank you in advance and sorry for the vague formulation (again)!