On the Grothendieck-Serre conjecture about principal bundles and its generalizations

Let $U$ be a regular connected affine semi-local scheme over field $k$. $G$ reductive group $U$. Assuming that has an appropriate parabolic subgroup scheme, we prove the following statement. Given $k$-scheme $W$, principal $G$-bundle $W\times_kU$ is trivial if it generic fiber of projection $W\times_kU\to U$. We also simplify proof Grothendieck-Serre conjecture: let A point generalize some other related results from simple simply-connected case to arbitrary schemes.