Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$, let $\pi:P\to M$ be a smooth principal right $G$-bundle, and let $\omega\in\Omega^1(P;\mathfrak{g})$ be a connection form. Let $\Omega\in\Omega^2(P;\mathfrak{g})$ be its curvature form. For $\xi\in\mathfrak{g}$, let $\xi_P\in\mathfrak{X}(P)$ denote the fundamental vector field generated by $\xi$ under the right action.

Then $\Omega$ is horizontal and $\operatorname{Ad}$-equivariant. Explicitly, for every $p\in P$, every $\xi\in\mathfrak{g}$, and every $X\in T_pP$,

\begin{align*} \Omega_p(\xi_P(p),X)=0 \end{align*}

and

\begin{align*} \Omega_p(X,\xi_P(p))=0. \end{align*}

Moreover, for every $g\in G$,

\begin{align*} (R_g)^*\Omega=\operatorname{Ad}_{g^{-1}}\Omega, \end{align*}

where $R_g:P\to P$ is the principal right action map $p\mapsto p\cdot g$.