Let $H$ and $G$ be Lie groups with identity elements $e_H$ and $e_G$, Lie algebras $\mathfrak h$ and $\mathfrak g$, exponential maps $\exp_H:\mathfrak h\to H$ and $\exp_G:\mathfrak g\to G$, and adjoint representations $\operatorname{Ad}^H:H\to GL(\mathfrak h)$ and $\operatorname{Ad}^G:G\to GL(\mathfrak g)$. Let $\varphi:H\to G$ be a smooth Lie [group homomorphism](/page/Group%20Homomorphism), and let $d\varphi_e:=d\varphi_{e_H}:\mathfrak h\to\mathfrak g$ be its derivative at the identity element of $H$. Let $M$ be a smooth manifold, let $\pi_Q:Q\to M$ be a smooth principal $H$-bundle with right action $R_h^Q:Q\to Q$, and assume that $Q$ admits a principal connection. Let $\pi_{\widetilde Q}:\widetilde Q:=Q\times_{\varphi}G\to M$ be the extension of structure group, where $Q\times_{\varphi}G$ is the quotient of $Q\times G$ by the right $H$-action $(q,g)\cdot h=(q h,\varphi(h)^{-1}g)$, with principal right $G$-action $[q,g]\cdot a=[q,ga]$.

For each integer $k\ge 0$, let $I^k(G)$ denote the [vector space](/page/Vector%20Space) of symmetric $k$-linear maps $P_0:\mathfrak g^k\to\mathbb R$ invariant under $\operatorname{Ad}^G$, with $I^0(G)=\mathbb R$, and set $I(G):=\bigoplus_{k\ge 0}I^k(G)$. For every homogeneous $P_0\in I^k(G)$, define $\varphi^*P_0\in I^k(H)$ by $(\varphi^*P_0)(X_1,\dots,X_k):=P_0(d\varphi_e(X_1),\dots,d\varphi_e(X_k))$ for all $X_1,\dots,X_k\in\mathfrak h$, and extend this definition degree by degree to $I(G)$.

Then, for every $P_0\in I(G)$, the Chern-Weil classes satisfy $\operatorname{cw}_{Q\times_{\varphi}G}(P_0)=\operatorname{cw}_Q(\varphi^*P_0)$ in $H^{\mathrm{even}}_{\mathrm{dR}}(M)$.