Let $G$ be a Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$, let $\pi:E\to M$ be a smooth right principal $G$-bundle over a smooth manifold $M$, and let $k\ge 1$ be an integer. Let
for every $g\in G$ and every $X_1,\dots,X_k\in\mathfrak g$. Let $\operatorname{ad}E:=E\times_{\operatorname{Ad}}\mathfrak g$ be the adjoint bundle. Extend $P$ to $\operatorname{ad}E$-valued differential forms by the Chern-Weil graded wedge convention: if $\alpha_i\in \Omega^{p_i}(M;\operatorname{ad}E)$, then
is obtained locally by wedging the form components and applying $P$ to the $\mathfrak g$-components, with the induced Koszul signs. In particular, for $\alpha\in\Omega^p(M;\operatorname{ad}E)$, write
Let $A_0,A_1\in \operatorname{Conn}(E)$ be smooth principal connections on $E$. Using the affine structure on $\operatorname{Conn}(E)$ modeled on $\Omega^1(M;\operatorname{ad}E)$, define