I am interested in considering vector bundles $E$ of rank $n$ over a base $X$ with fiber $V \cong \mathbb{C}^{n}$ such that the structure group $G \subseteq \text{GL}_{n}(\mathbb{C})$ is possibly a Lie subgroup of $\text{GL}_{n}(\mathbb{C})$, with Lie algebra $\mathfrak{g}$. Here are a few important examples:
$G = \text{GL}_{n}(\mathbb{C})$, and $\mathfrak{g} = \mathfrak{gl}_{n}(\mathbb{C})$ if and only if $E$ is a complex vector bundle with no additional structure.
$G = \text{SL}_{n}(\mathbb{C})$, and $\mathfrak{g} = \mathfrak{sl}_{n}(\mathbb{C})$, if and only if $E$ is a complex vector bundle with trivial determinant $\bigwedge^{n} E \cong \mathcal{O}_{X}$.
$G=\text{U}(n)$ and $\mathfrak{g}=\mathfrak{u}_{n}$ if and only if $E$ is a complex vector bundle with a Hermitian metric.
$G=\text{SU}(n)$ and $\mathfrak{g}=\mathfrak{su}_{n}$ if and only if $E$ is a complex vector bundle with a Hermitian metric and $\bigwedge^{n} E \cong \mathcal{O}_{X}$.
Now, where I'm slightly confused is when we start studying more "gauge-theoretic" objects on and surrounding the bundle, i.e connections, curvatures, group of gauge transformations etc.
We can consider the bundle of Lie algebras associated to the adjoint representation of $G$ which we denote $\mathfrak{g}_{E} \subseteq \text{End}E$. Basically, we should have (differences between) connections lying in $\Omega^{1}(X, \mathfrak{g}_{E})$ and curvature two-forms lying in $\Omega^{2}(X, \mathfrak{g}_{E})$.
My question is the following: there is some sort of "equation" or "form" responsible for the restriction of the structure group from $\text{GL}(V)$ to $G$. Is there a clean way to write out how this form induces a constraint which the connections and curvatures must satisfy in order to be valued in $\mathfrak{g}_{E}$? Is this also responsible for cutting $\mathfrak{g}_{E}$ itself out of $\text{End}E$?
As an example of what I'm referring to above is that a Hermitian connection on a Hermitian vector bundle should satisfy the additional condition that $dh(s,t) = (\nabla s, t) + (s, \nabla t)$.
Also, I think what I've written here is slightly confusing/misleading because it's unclear if $E$ is a complex vector bundle without any other structure or if $E$ refers to a complex vector bundle with any extra structure described above.
I think I understand mostly what's going on here, but my head starts spinning when trying to write it up in a clean, consistent, and general way. Thanks for any help!