Let $\phi : Y \to X$ be an $X$-torsor under $G_X = G \times_k X$, where $Y,X$ are $k$-schemes and $G$ is an algebraic $k$-group. Let $U \subseteq X$ be an open subset, and let $W = \phi^{-1}(U)$. I'm wondering the following:
Is $W \to U$ a $G_U$-torsor?
If the answer to the above is yes, I have a further question:
does twisting $ Y \to X$ by say $\sigma \in Z^1(k,G)$ induce a twisting of $W \to U$ by $\sigma$?