Let $\pi:P\to M$ be a principal $G$-bundle over a smooth manifold $M$, let $\omega\in\Omega^1(P;\mathfrak g)$ be a principal connection form, and let $B:V\times V\to\mathbb R$ be a nondegenerate [bilinear form](/page/Bilinear%20Form) preserved by $\rho:G\to GL(V)$. Form $E=P\times_G V$ using the convention $[pg,v]=[p,\rho(g)v]$, and let $B_E$ be the associated fibrewise bilinear form. Then the connection on $E$ induced by $\omega$ satisfies

\begin{align*} d(B_E(s,t))=B_E(\nabla s,t)+B_E(s,\nabla t) \end{align*}

for all $s,t\in\Gamma(E)$.