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...

I wonder that how to reduce the proof to the case when $G$ is semisimple of adjoint type, so I can use Iwahori's theorem. Thank for any help.