Question on geometric invariant theory

Question

I'm reading Geometric Invariant Theory by Mumford, and confuse about the Proposition 2.4 on P54.

It states that:

Let $G$ be a reductive group, act on an algebraic scheme. Then the action of $G$ on $X$ is proper if and only if for every non-trivial 1-PS $\lambda: \mathbb{G}_m\to G$, the induced action of $\mathbb{G}_m$ on $X$ is proper.

Here the action is proper means $(\sigma, p_2):G\times X\to X\times X$ is proper as morphism.

I get confused since in the proof of this proposition, the book use a theorem of Iwahori on P52, which is stated for $G$ semisimple of adjoint type... Mumford use a similar argument on P53, which also just require $G$ to be reductive....

I wonder that how to reduce these proofs to the case when $G$ is semisimple of adjoint type, so I can use Iwahori's theorem. I try to pass to the case when $G= G/C(G)$, but I can't find a natural action of $G/C(G)$ on $X$.

Thanks for any help.

Answer 1

I believe this is addressed on page 52, underneath the statement of Iwahori's theorem. He provides an argument for why Iwahori's result can be strengthened to G reductive by considering $G \rightarrow G'$ where $G $ is reductive and $G'$ is the associated adjoint group.