Let $P\to M$ be a smooth principal $G$-bundle, and let $\operatorname{Gau}(P)$ denote the group of vertical principal-bundle automorphisms of $P$. There is a natural group isomorphism
where $\Gamma^\infty(\operatorname{Ad}(P))$ denotes the smooth sections of the adjoint group bundle, with fibrewise multiplication. The section corresponding to $\Phi(p)=p\gamma(p)$ is $x\mapsto [p,\gamma(p)]$ for any $p\in P_x$.