Connection and reduction of the structure group

Question

I am writing a memoir about gauge theory. I have trouble with a small proof which should be simple and have the feeling that I am missing something obvious. I want to show that the set of connections on a vector bundle $E \to M$ of rank $p$ with a reduction of the the structure group to $H$ (a closed subset of $GL_p(\mathbb{R})$ ie a Lie group) is non-empty. Obviously, I want to glue via a partition of unity local trivial connections $\nabla^{i}$ over $U_i \subset M$ an open trivializing set. But I need first to show that a trivial local connection $d$ for the trivialization $\Phi_i$, ie the 1-form with matrix values of $\nabla^i$ is null in this trivialization, is compatible with the reduction of the structure group to $H$, which means its matrix belongs to $\mathfrak{h} = Lie(H)$ for any trivialization compatible with the reduction of the structure group. But if $\Phi_j$ is such a trivialization, with the transition fonctions $\phi_{ji,x} \in H$, the connection matrix in $\Phi_j$ is
$\Gamma^{\nabla^i}_{\Phi_j}(x) = - (d_x \phi_{ji}) \phi_{ji,x}^{-1}$ and it doesn't seems that belongs to $\mathfrak{h}$.
What am I doing wrong ?
Thanks for your answers...