I'm in particular interested in understanding Grothendieck's argument for this in SGA 1 (page 232 in http://arxiv.org/pdf/math/0206203v2.pdf)

Let $G$ be $\text{GL}_n$ over a scheme $S$ for some integer $n$, and let $P/S$ be a principal $G$-bundle. Then we know that there is an fpqc morphism $S'\rightarrow S$ such that $P' := P\times_S S'$ is isomorphic to $G' := G\times_S S'$ over $S'$.

He claims in his proof that the statement follows from noting that $G(T) = \text{Aut}(\mathcal{O}_T^n)$ (for any $S$-scheme $T$), and that fpqc morphisms are morphisms of effective descent in the category of locally free $\mathcal{O}_T$-modules of rank $n$.

I guess his argument must rely on something special about automorphism group schemes which I'm missing. I admit I haven't read the entirety of his prior chapters on descent. Everything I know about descent pretty much comes from reading the stacks project.

I'd like to know how Grothendieck envisaged his argument would go, though I'd also appreciate any other argument proving this fact or relevant references.

thanks

- will