The universal connection for principal bundles over homogeneous spaces and twistor space of coadjoint orbits

Given a holomorphic principal bundle $Q, longrightarrow, X$, the universal space of holomorphic connections is a torsor $C_1(Q)$ for $text{ad} Q otimes T^*X$ such that the pullback of $Q$ to $C_1(Q)$ has a tautological holomorphic connection. When $X,=, G/P$, where $P$ is a parabolic subgroup of a complex simple group $G$, and $Q$ is the frame bundle of an ample line bundle, we show that $C_1(Q)$ may be identified with $G/L$, where $L, subset, P$ is a Levi factor. We use this identification to construct the twistor space associated to a natural hyper-Kahler metric on $T^*(G/P)$, recovering Biquards description of this twistor space, but employing only finite-dimensional, Lie-theoretic means.