I'm reading this notes, and I have a question about the construction on page 48.
Let $M=S^{1}\times \mathbb{R}^{3}$ and $E\rightarrow M$ a vector bundle over $M$. If $A$ is the $1$-form matrix associeted to a connection on $E\rightarrow M$, looking it on $I\times \mathbb{R}^{3}$, we can write: $$ A=A_{0}dt+A_1dx_1+A_2dx_2+A_3dx_3. $$
Assume that $A_i$ only depend on $t$, for $i=0, \dots,3$. My question is:
Why $gAg^{-1}+dg\cdot g^{-1}$ still depending only on $t$, for all $g\in \mathcal{G}$?