Let $U$ be a $k$-scheme, where $k$ is a field. Let $G$ be a smooth affine $k$-group. Recall that a principal $G$-bundle over $U$ is a smooth surjective $U$-scheme $E$ with an action of $G$ on $E$ such that the action commutes with the projection to $U$ and the obvious morphism $G\times E\to E\times_UE$ is an isomorphism.

It is easy to see that every principal $G$-bundle is locally trivial in etale topology. However Serre in his *Espaces fibres algebriques* required a stronger condition called *local isotriviality*: for all $u\in U$ there is a a *finite* etale morphism $T'\to T$, where $T$ is a Zariski neighborhood of $u$ such that the pull-back of $E$ to $T'$ is trivial. I wonder if these definitions are known to coincide (I think I can prove it in some generality but the proof is not trivial).

Sketch of a proof. Let us embed $G$ into $GL(n)$. Consider the associated space $E'=E\times^GGL(n)$. It is a scheme because $GL(n)$ is affine and affine schemes can be glued in any reasonable topology. Moreover, it is a principal $GL(n)$-bundle, so passing to a Zariski cover we can assume it is trivial. The original bundle can be obtained from E' via reduction of the structure group, that is, it is a pull-back of the principal $G$-bundle $GL(n)\to GL(n)/G$. It remains to use the isotriviality of the latter bundle.