I'm looking for a reference for the following proposition:
Let $G$ be a Lie group and $H$ a (closed) Lie subgroup of $G$. Let $E \to B$ be a principal $G$-bundle. Then reductions of the structure group from $G$ to $H$ of $E$ are in bijective correspondence with sections of the $G/H$-bundle $E/H \to B$.
I want to know all about the details of the argument, so a more comprehensive source is preferred.