Currently reading a paper on finding solutions to the Yang Mills equation
$D^*\Omega=0$,
where $\Omega$ is the curvature and $D^*$ is the formal adjoint of the exterior covariant derivative $D$. The paper expresses the quantity $D^*\Omega$ locally as follows:
For a local orthonormal frame field $(e_1,\dots,e_m)$ on a Riemannian manifold $(M,h)$, they define $(e_i^*)_{i=1}^m$ to be the horizontal lifts to the bundle $P$ relative to the connection $\omega$. Then \begin{align} D^*\Omega&=-\sum_{i,j}^m(\nabla_{e_i}\Omega)(e_i^*,e_j^*)\pi^*\omega_j \end{align} where $\{\omega_i\}$ is a system of 1-forms on $M$ dual to $\{e_i\}$ and $\nabla$ is the Levi Civita connection of $(M,h)$.
I am just wondering if there is a quick way to verify that this is indeed the formula for the formal adjoint of the curvature and how would you go about doing so?