This is from Chapter 3 of Uhlenbeck and Freed's Instantons and Four-Manifolds.
As a quick setup of the relevant notation, we fix a complex rank $2$ bundle $\eta$ with gauge group $SU(2)$ and examine the space of connections. Let $P$ be the associated frame bundle, itself a principal $SU(2)$-bundle.
The gauge transformations of $\eta$ are then just the bundle automorphisms of $P$. The connections on $\eta$ can be identified with the affine space of $1$-forms on the adjoint bundle $\text{ad} P = P \times_{SU(2)} \mathfrak{su}(2)$, written as $\Omega^1(\text{ad} P)$.
Furthermore, any connection $D$ also has an associated covariant derivative on $\text{ad} P$, so it is realized as a differential operator $D: \Omega^0(\text{ad} P) \to \Omega^1(\text{ad} P)$.
The group of gauge transformations acts on a connection $D$ by pullback, i.e. $s^*(D)(\sigma) = s^{-1}D(s \sigma)$ where $s$ acts on $\sigma$ by conjugation.
We can also realize the group of gauge transformations as sections of the bundle $\text{Ad} P = P \times_{SU(2)} SU(2)$, where the group action is conjugation. The Lie algebra of this group is therefore clearly $\Omega^0(\text{ad} P)$.
Given the action of the gauge group, the authors then claim the infinitesmal action of a Lie algebra element $u \in \Omega^0(\text{ad} P)$ on a connection $D \in \Omega^1(\text{ad} P)$ is equal to $Du \in \Omega^1(\text{ad} P)$.
I wrote out the definition of infinitesmal action, but I am not sure how exactly to evaluate the relevant derivative. Can anyone provide a hint/insight into this calculation?