It seems implicitly written, on the Wikipedia page on $G$-structures, that if a (closed) subgroup $H$ is a deformation retract of a Lie group $G$, then given a $G$-principal bundle $P$, there always exists a reduction of the structure group of $P$ from $G$ to $H$.
Where can I find a proof of this fact?
And perhaps more importantly is there a proof that is readable by non-experts in algebraic topology that I could use as a reference in an article?