First, why the vector bundle, structured from principal bundle, has structure group $G$ ?
Second, how to prove the principal $G$-bundle, structured from vector bundle, is principal bundle according the Definition 2.3.3 ?
Picture below is from the 65 of Jost's Riemannian Geometry and Geometric Analysis. In fact ,I think the definition 2.3.3 is not a good definition.
