Let $Bun(X)$ denote the set of equivalence classes of complex rank 2 vector bundles with a reduction of its structure group to $\mathbb{H}^*$. How can I proof that there is a bijection between $Bun(X)$ and $ [X,B\mathbb{H}^*]$?
I know that there is a bijection between the set of equivalence classes of $\mathbb{H}^*$-bundles $Prin_{\mathbb{H}^*}(X)$ and $[X,B\mathbb{H}^*]$, so I'm trying to find some bijection between $Bun(X)$ and $Prin_{\mathbb{H}^*}(X)$.
Any help is appreciated. Thanks in advance.