Is there a natural construction of $G$-connection $\omega^*$ on the frame bundle $P$ given a covariant derivative $\nabla$ defined on a vector bundle $\pi_E:E\to M$?(The structure group $G$ here is then general linear group on vector bundle fiber)
Since $\nabla$ differs from $d$ by a form $\omega\in \Omega^1(M,End(E)=ad(P))$, we can pullback $\omega$ to $\omega^*\in\Omega^1(P,ad(P))$ by projection $\pi_P:P\to M$. Now,given a basis $e_1,...,e_n$ of lie algebra $end(E)$, we can find the associated fundamental vector fields $e_i^*$ and the dual 1-forms $\epsilon_i^*$. Define an $end(E)-$ valued 1-form $\omega_0= \sum \epsilon_i\otimes e_i$ on $P$ then $\omega_0$ will give back lie algebra element from its fundamental vector field. I then think the $G-$equivariant principal connection form on $P$ must be $\omega_0+\omega^*$ where $\omega^*=0$ corresponding to flat-connection. However, there seems to be the problem of showing that the said $\omega^*$ is $G-$equivariant.
Am I on the right track, or is there more natural ways to construct the principal connection on $P$ from $\omega$? I'll be delighted if someone could help me on this.
Thanks.