Let $X$ be a variety and $E$ be a principal $G$ bundles, where $G$ is a semisimple group. Is there a variety $f: \tilde{X}\rightarrow X$ such that $f^\*E$ is trivial $G$ bundle?



Global trivialization of a Principal G bundle
Yes, a principal $G$-bundle is trivial if and only if it has a section, and if you take $\tilde{X} = E$, then $f^\*E = E \times\_X E$ has a section given by the diagonal map $E \to E \times\_X E$.


