Suppose we consider $\operatorname{ad}P_G \to T^k$ as the associated adjoint bundle (maybe this is not the correct name, but I just mean with the associated vector bundle ${\rm Lie}G$ as standard fiber in the adjoint rep.) over $T^k$, which is parameterized by standard coordinates $(\varphi_1, ..., \varphi_k)$.
Let vector field $R$ be $\partial_{\varphi_1}$ (or any vector field that generated cycles on $T^k$). Finally denote $A$ as connection on $\operatorname{ad} P_G$.
My questions are:
Can I always choose a gauge such that $\iota_R A = 0$ or $\mathcal{L}_R A = 0$?
If not, are there known gauge choices that are optimized in the presence of a vector field $R$ (I am guessing the space of connection $\mathcal{A}$ is decomposed into subspaces, and in each subspace has its own "optimal" gauge.)?
I think the answer to (1) is probably NO, since counter examples ($A$ with non-zero holonomy around the cycle of $R$, which cannot be made into $\iota_R A = 0$ via gauge transformations) are there; but I am not sure if $\mathcal{L}_R A = 0$ is also ruled out somehow.