Let $S$ be a scheme over a field $k$, and let $G$ be a reductive group scheme over $S$. Let us call it *trivial*, if it is a pull-back of a group scheme over $k$ via the structure morphism $S\to k$. Is it always true that $G$ becomes trivial after a certain etale base change $S'\to S$? I am willing to assume that $S$ is smooth if needed.



Is every reductive group scheme etale locally trivial?
Reductive groups schemes over $S$ are classified by $H^1\_{fpqc}(S,Aut\_G)$, see SGA 3 Exp. XXIV.


