I defined the connection 1-form on a principal bundle $\mathcal{P}=(P,M,\pi;G)$ as the $\mathfrak{g}$-valued 1-form
$ \bar{\omega_{p}}=\left( \theta^{A}_{(L)}(p)+\mathrm{Ad}^{A}_{B}(g^{-1})\omega^{B}_{\mu}\mathrm{d}x^{\mu} \right) \otimes T_{A} $
where $\mathfrak{g}$ is the Lie algebra of the group G with basis $\{ T_{A} \}$, $\theta^{A}_{(L)}=L^{A}_{a}(g)\mathrm{d}g^{a}$ is the local basis of left-invariant 1-forms on the bundle and $p=[x,g]_{(\alpha)}$ is a point on the bundle with respect to a local trivialization. Then, I defined the exterior covariant derivative (ECD) of $\mathfrak{g}$-valued $k$-forms with respect of the connection $\omega$ as the $\mathfrak{g}$-valued $(k+1)$-form given by:
$ \mathrm{D}_{\omega}\theta \left( \Xi^{(1)},...,\Xi^{(k+1)}\right) = \mathrm{d}\theta\left(\Xi^{(1)}_{(H)},...,\Xi^{(k+1)}_{(H)}\right)$
where $\Xi^{(1)}_{(H)},...,\Xi^{(k+1)}_{(H)}$ are the horizontal parts of the vector fields $\Xi^{(1)},...,\Xi^{(k+1)}$ with respect of said connection. This means that in order to find the ECD of a form, one should first compute the usual differential of said form, and then contract it with the horizontal parts of $k+1$ vector fields via the connection.
My goal is to calculate the exterior covariant derivative of the connection 1-form $\bar{\omega}$, which according to the book I'm using as reference ("Natural and Gauge Natural Formalism for Classical Field Theories" by Lorenzo Fatibene and Mauro Francaviglia), it should have this form:
$ \mathrm{D}_{\omega}\bar{\omega}=\frac{1}{2}\mathrm{Ad}^{A}_{D}(g^{-1})\left( \partial_{\mu}\omega^{D}_{\nu} - \partial_{\nu}\omega^{D}_{\mu}+c^{D}_{\cdot BC}\omega^{B}_{\mu}\omega^{C}_{\nu}\right)\mathrm{d}x^{\mu}\wedge\mathrm{d}x^{\nu}\otimes T_{A} $
and it is referred to as the curvature 2-form, where $c^{D}_{\cdot BC}$ is the structure costant of the Lie algebra $\mathfrak{g}$ given by $-c^{D}_{\cdot BC} \rho_{D} =[\rho_{B},\rho_{C}]$, with $\rho_{A}$ being a right-invariant field on the principal bundle.
The book omits many details and I have tried countless times to compute the ECD, but I eventually got stuck everytime, and I don't know what to do. I know that you can use known formulas to relate the differential of the adjoint matrix Ad to structure constants, but they still don't get me anywhere. I very much hope somebody else has gone through this already and can give me a hand, or at least some hints on how to structure my computation.