Let $G$ be a Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$, let $\pi:P\to M$ be a smooth principal $G$-bundle over a smooth manifold $M$, let $k\ge 1$, and let $q:\mathfrak g^k\to\mathbb R$ be a symmetric $k$-[linear map](/page/Linear%20Map) that is invariant under the adjoint action of $G$. For every principal connection $A\in\Omega^1(P;\mathfrak g)$ on $P$ with curvature $F_A\in \Omega^2(P;\mathfrak g)$, let $q(F_A)\in \Omega^{2k}(M)$ denote the unique descended form satisfying

\begin{align*} \pi^*q(F_A)=q(F_A,\dots,F_A)\in\Omega^{2k}(P), \end{align*}

where the right-hand side is formed by the standard Chern-Weil extension of $q$ to $\mathfrak g$-valued differential forms. Then $q(F_A)$ is closed. Moreover, if $A_0$ and $A_1$ are two principal connections on $P$, then

\begin{align*} q(F_{A_1})-q(F_{A_0}) \end{align*}

is exact in $\Omega^{2k}(M)$. Consequently the de Rham class

\begin{align*} [q(F_A)]\in H^{2k}_{\mathrm{dR}}(M) \end{align*}

is independent of the chosen principal connection $A$.