Let $G$ be a Lie group with Lie algebra $\mathfrak g$, let $H \le G$ be a closed Lie subgroup with Lie algebra $\mathfrak h$, and let $\pi: P \to M$ be a smooth principal $G$-bundle. Let $Q \subset P$ be a smooth principal $H$-subbundle over $M$, so that $\pi|_Q: Q \to M$ is a principal $H$-bundle and the inclusion $Q \hookrightarrow P$ covers $\operatorname{id}_M$.

Let $\omega \in \Omega^1(P;\mathfrak g)$ be a principal $G$-connection form, and define its horizontal distribution by

\begin{align*} \mathcal H_q^\omega := \ker(\omega_q) \subset T_qP \end{align*}

for each $q \in P$. Then the reduction $Q$ is preserved by $\omega$, meaning

\begin{align*} \mathcal H_q^\omega \subset T_qQ \qquad \text{for every } q \in Q, \end{align*}

if and only if the restricted form

\begin{align*} \omega_Q := \omega|_{TQ} \in \Omega^1(Q;\mathfrak g) \end{align*}

takes values in $\mathfrak h$ and, viewed as an element of $\Omega^1(Q;\mathfrak h)$, is a principal $H$-connection form on $Q$.