Let $P\rightarrow M$ be a principal $G$-bundle with a connection 1-form $\omega$. In a local trivialisation $\tau_U \colon U\rightarrow P_U$ ($U \subset M$) we can pull the connection back to the base manifold. $$\omega_U=\tau_U^*\omega=\omega^i \otimes e_i \, .$$ Now consider the adjoint bundle $\text{ad} P$ of $P$. The principal connection induces a covariant derivative on $ad P$. In the same trivialising neighbourhood, it is given by the formula $$\nabla_X s =[\omega_U(X),s]+ds(X)$$
I am interested in the other direction. Suppose we start with a covariant derivative $\nabla=d+A$ on $\text{ad} P$. What conditions should $A$ satisfy so that there exists $\omega$ on $P$ that induces $\nabla$?
In particular, I am interested in the case when $G$ is compact, connected, semisimple. I have very little experience with Lie algebras, but I believe this should help somehow.