Natural Action of Killing vector fields on space of connections

I try to understand some mathematical aspects of supersymmetric Yang Mills theory following the book "Quantum fields and strings - a course for mathematicians". In this context the following question came up:

Given a (pseudo)-Riemannian manifold $(M,g)$ with Killing vector fields $Kill(M,g)$ and a $G-$principal bundle $P \rightarrow M$, is there a natural action $\rho$ of $Kill(M,g)$ on the space of connections of $P$, or after fixing one connection, on the space $\Omega^1(M,Ad(P))$ ?

The book says that infinitesimal symmetries act on connections by "Lie derivative by the horizontal lift", so given a connection $A \in \Omega^1(P,\mathfrak{g})$, I would set $\rho_1(X)(A):=A+L_{X^*}A$ for $X \in Kill(M,g)$, which can easily be shown to be a connection again. However, I could not prove that this is compatible with the Lie bracket and I could not show that it leads to a representation of $Kill(M,g)$ on $\Omega^1(M,Ad(P))$.

In Physics literature I often found local expressions for the action of infinitesimal symmetries on local connection 1-forms. When translated into the above notation they reads as $(\rho_2(X)A)^s = L_X (A^s)$, where $A^s \in \Omega^1(U,\mathfrak{g})$ is the local connection 1-form wrt. a section $s:U \rightarrow P$. However, I dont see that the LHS of this equation has the right transformation behaviour under a change of $s$ in order to give a well defined connection on $M$.

Did I understand something wrong here ?