Let $Q \subset P$ be an $H$-reduction of a principal $G$-bundle, and suppose $\mathfrak g=\mathfrak h \oplus \mathfrak m$ is a reductive splitting. If $\omega \in \Omega^1(P;\mathfrak g)$ is a principal $G$-connection form, then on $Q$ there is a unique decomposition
where $\omega_{\mathfrak h} \in \Omega^1(Q;\mathfrak h)$ is a principal $H$-connection form and $\omega_{\mathfrak m} \in \Omega^1(Q;\mathfrak m)$ vanishes on the vertical tangent bundle of $Q \to M$ and is $H$-equivariant. Moreover, $Q$ is preserved by $\omega$ if and only if $\omega_{\mathfrak m}=0$.