I'm working within the jet-formulation espoused by Saunders in "The Geometry of Jet Bundles" and am struggling to prove the stated result. I would like to stay in this context and understand the result in the following terms.
The context is this. Given a principal $G$-bundle $\pi:P\rightarrow M$ over a manifold $M$ for some Lie group $G$, the first jet prolongation of $\pi$ is the manifold $J^1P$, whose points are equivalence classes $j^1_ps$ of local sections $s$ defined in some neighbourhood $U$ of $p$. Two local sections $s$, $t$ at $p$ are equivalent if both conditions
$$s(p)=t(p),\qquad \frac{\partial s^a}{\partial x^i}=\frac{\partial t^a}{\partial x^i}$$
hold for some fibred cordinates $(x^i,u^a)$ at $s(p)$. The first jet manifold fibres over $P$, with the projection $\pi_{1,0}:J^1P\rightarrow P$, $j^1_ps\mapsto s(p)$. The right $G$-action on $P$ extends to one on $J^1P$, given explicitly by $(j^1ps)\cdot g=j^1_p(s\cdot g)$, and defined thus the map $\pi_{1,0}$ is $G$-equivariant.
There is a contact map $\lambda:J^1P\rightarrow Hom(\pi^*TM,TP)$ as well as a complementary map $\theta:J^1P\rightarrow Hom(TP,VP)$. These are bundle maps over $P$. They are embeddings and enjoy certain equivariance properties with respect to the $G$-actions on all spaces. Explicitly they are defined by
$$\lambda(j^1_ps)=T_ps,\qquad \theta(j^1_ps)=1-T_ps\circ T_{s(p)}\pi.$$
In this context a connection on $\pi$ is a section of $\pi_{1,0}$, given by a map $\omega:P\rightarrow J^1P$. A connection $\omega$ is said to be principal if it is equivariant with respect to the right $G$-action.
The standard result is that the "difference" of two connections descends to give a well-defined 1-form on $M$ with values in the Lie algebra $\mathfrak{g}$. This statement should be understood in terms of (one of) the embeddings $\lambda$, $\theta$. However I cannot prove it. My attempt below yields an $ad_g$-dependence on my choice of lifting vector fields of which I cannot rid myself.
I proceed as follows. I assume given two principal connections $\omega_1,\omega_2:P\rightarrow J^1P$. At an arbitrary point $p_0\in M$ I choose a local section $s$ defined in some neighbourhood $U$ of $p_0$ and define a $\mathfrak{g}$-valued 1-form $A^s_U$ on $U$ by setting
$$A^s_U(X)(p)=T_{s(p)}\hat s\circ(\theta\circ\omega_2(s(p))-\theta\circ\omega_1(s(p)))\circ T_ps(X_p)$$
where $X\in\mathcal{X}(U)$ is a vector field on $U$ and $p\in U$. Here $\hat s:P|_U\rightarrow G$ is the local trivialisation of $P$ determined by $s$. That is, for $e\in P|_U$, it is defined by $e=s(\pi(p))\cdot \hat s(e)$.
The tangent map $T_ps$ lifts $X$ to $TP$. Then each $\theta(\omega_i(s(p))\in Hom(T_{s(p)}P,V_{s(p)}P)$ sends it to the vertical subspace. Finally $\hat s(s(p))=1_G$, so the tangent map $T_{s(p)}\hat s$ takes values in $T_1G\cong\mathfrak{g}$.
I need to show that the choice of local section $s$ is of no consequence, so that $A_U^s=A_U$. This will then allow for the various $A_U$'s to be patched together as $U$ ranges over an open cover of $M$.
My difficulty is that if I choose a different section $t:U\rightarrow P|_U$ (defined without loss of generality on the same neighbourhood of $p_0$) then over $U$ it is related to $s$ by $t(p)=s(p)\cdot f(p)$, for some map $f:U\rightarrow G$, and I get
$$A^t_U=ad_{f(p)^{-1}}\circ A^s_U$$
where $ad_{f(p)^{-1}}:\mathfrak{g}\rightarrow \mathfrak{g}$ is the adjoint action of $G$ on its Lie algebra. I can't rid myself of this conjugation. Where am I going wrong?