Let $E \to (M,g)$ be a vector bundle over $M$ a compact riemannian manifold and consider the structural group $G$ of bundle. We can define a linear connection over $E$ as a map $\nabla: \Gamma(E) \to \Omega^{1}(M,E)$ given by $\nabla: D+w$, where $w \in \Omega(M,Lie(G))$ and $D$ is a differential operator which acts on $C^{\infty}(M)$. If we consider the line bundle $L \to (M,g)$ with structural group $U(1)$, we have that the $u(1)$-valued 1-form $w$ this bundle can be given by $i\alpha$, where $\alpha \in \mathbb{R}$, thus we can write the space of u(1)-valued connection 1-form as $i\Omega(M,\mathbb{R})$.
My question: I would like to obtain a result to the structural group of SU(2). I would like to know if I could obtain some result to $Im(\mathbb{H}) \cong su(2)$-valued connection 1-form. Because I would like to write this 1-form as a product of quartenions.