Let $G$ be a Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$, adjoint representation $\operatorname{Ad}:G\to GL(\mathfrak g)$, and Lie bracket $[\cdot,\cdot]:\mathfrak g\times\mathfrak g\to\mathfrak g$. Let $M$ be a smooth manifold, and let $\pi:P\to M$ be a smooth principal $G$-bundle with right action maps $R_g:P\to P$. Let $A_0,A_1\in\Omega^1(P;\mathfrak g)$ be principal connection forms on $P$, and for each $i\in\{0,1\}$ let the curvature form $F_{A_i}\in\Omega^2(P;\mathfrak g)$ be defined by \begin{align*} F_{A_i}=dA_i+\frac{1}{2}[A_i\wedge A_i]. \end{align*} Let $k\ge 0$, and let $P_0\in I^k(G)$, meaning that for $k=0$ the symbol $P_0$ denotes a real constant and for $k\ge 1$ it denotes a symmetric $k$-[linear map](/page/Linear%20Map) $P_0:\mathfrak g^k\to\mathbb R$ satisfying \begin{align*} P_0(\operatorname{Ad}_g X_1,\dots,\operatorname{Ad}_g X_k)=P_0(X_1,\dots,X_k) \end{align*} for every $g\in G$ and every $X_1,\dots,X_k\in\mathfrak g$. For $k\ge 1$, let $P_0(F_{A_i})\in\Omega^{2k}(P)$ denote the real-valued differential form obtained by wedging the form components of $k$ copies of $F_{A_i}$ and applying $P_0$ to their $\mathfrak g$-components; for $k=0$, set $P_0(F_{A_i})=P_0$. For each $i\in\{0,1\}$, let $P_0(F_{A_i})_M\in\Omega^{2k}(M)$ denote the unique descended Chern-Weil form satisfying \begin{align*} \pi^*P_0(F_{A_i})_M=P_0(F_{A_i}). \end{align*} Then the closed forms $P_0(F_{A_0})_M$ and $P_0(F_{A_1})_M$ represent the same de Rham cohomology class: \begin{align*} [P_0(F_{A_0})_M]=[P_0(F_{A_1})_M]\in H^{2k}_{\mathrm{dR}}(M). \end{align*}