I am aware of gauge transformations and covariant derivatives as understood in Quantum Field Theory and I am also familiar with deRham derivative for vector valued differential forms.
I thinking of the gauge field A of the gauge group G as a Lie(G) valued 1-form on the manifold.
But I can't see why under a gauge transformation on A by an element $g\in G$ amounts to the following change, $A \mapsto A^g = gAg^{-1} -dgg^{-1}$ (if G is thought of as a matrix Lie Group) or in general $A_g = Ad(g)A + g^* \omega$ (where $\omega$ is the left invariant Maurer-Cartan form on G and I guess $g^*$ is pull-back of $\omega$ along left translation map by $g$).
Curvature is defined as $F = dA + \frac{1}{2}[A,A]$ and using this one wants to now see why does $F \mapsto F_g = gFg^{-1}$.
Firstly is the expression for $A_g$ a definition or is there a derivation for that?
When I try proving this (assuming matrix Lie groups) I am getting stuck in multiple places like what is $dA_g$ ?
I would be happy if someone can explain the explicit calculations and/or give a reference where such things are explained. Usual books which explain differential forms or connections on principal bundles don't seem to help with such calculations.