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In a paper the principal $Sp(1)$-bundle $P$ over $S^4$ is introduced as follows: let $Sp(1)\times Sp(1)\hookrightarrow Sp(2) \xrightarrow{\pi} S^4$ be the spin structure on $S^4 $. The principal bundle $P$ is then defined as the associated $Sp(1)$ bundle \begin{equation} P= Sp(2) \times_{\rho} Sp(1), \end{equation} where $\rho: Sp(1)\times Sp(1) \rightarrow Sp(1) $ is simply projection onto the first $Sp(1)$ factor followed by multiplication in $Sp(1)$.

My question is: is that a general construction working for some classes of spaces or an ad hoc procedure which happens to work in this case? In the first case where can I find some references about it?

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