Let $G$ be a Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$, let $M$ be a smooth manifold, let $\pi:P\to M$ be a smooth principal right $G$-bundle, and let $k\ge 0$. Let
be an $\operatorname{Ad}$-invariant symmetric $k$-linear form on $\mathfrak g$, with the convention $I^0(G)=\mathbb R$. If $A_0,A_1\in\Omega^1(P;\mathfrak g)$ are principal connection forms with curvature forms $F_{A_0}$ and $F_{A_1}$, and if $P_0(F_{A_i})_M\in\Omega^{2k}(M)$ denotes the descended Chern-Weil form on $M$, then