Let $G$ be a Lie group, let $H \subset G$ be an embedded Lie subgroup, and let $\pi: P \to M$ be a smooth principal right $G$-bundle over a smooth manifold $M$. Suppose $P$ is represented on an open cover $(U_i)_{i \in I}$ by smooth transition functions $g_{ij}: U_i \cap U_j \to G$.

Then $P$ admits an $H$-reduction, meaning a smooth principal right $H$-bundle $\rho: Q \to M$ together with an $H$-equivariant smooth embedding $\iota: Q \hookrightarrow P$ over $M$ such that the induced extension-of-structure-group map $Q \times_H G \to P$ is an isomorphism of principal right $G$-bundles, if and only if, after passing to a refinement of the cover and replacing the local trivialisations of $P$ if necessary, the corresponding transition functions take values in $H$.

Equivalently, $P$ admits an $H$-reduction if and only if there is an open cover $(V_a)_{a \in A}$ of $M$ and smooth local sections $s_a: V_a \to P$ such that, for every $a,b \in A$, the unique transition map $h_{ab}: V_a \cap V_b \to G$ defined by

\begin{align*} s_b(x) = s_a(x) h_{ab}(x) \end{align*}

has image contained in $H$.