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, and let $A\in \Omega^1(P;\mathfrak g)$ be a principal connection with curvature form $F_A\in \Omega^2(P;\mathfrak g)$. Let $k\ge 0$, and let $P_0\in I^k(G)$, interpreted for $k\ge 1$ as a symmetric $k$-[linear map](/page/Linear%20Map) $P_0:\mathfrak g^k\to \mathbb R$ invariant under the adjoint action of $G$. Suppose $P_0(F_A)_M\in \Omega^{2k}(M)$ denotes the descended Chern-Weil form determined by

\begin{align*} \pi^*P_0(F_A)_M=P_0(F_A,\dots,F_A). \end{align*}

Then $P_0(F_A)_M$ is closed:

\begin{align*} dP_0(F_A)_M=0. \end{align*}