Let $\pi : E\to B$ be a fiber bundle with fiber $F$ over a finite complex $B$ whose structure group is a compact Lie group $G$. How can we determine the principal $G$-bundle associated to $\pi$? For example if we consider the bundle $S({\gamma}_{1}\times {\gamma}_{1})\to {BO}_{1}\times {BO}_{1}$ where ${\gamma}_{1}\to {BO}_{1}$ is the $1$-dimensional universal vector bundle over ${BO}_{1}$ and $S({\gamma}_{1}\times {\gamma}_{1})$ is the sphere bundle of ${\gamma}_{1}\times {\gamma}_{1}$ over ${BO}_{1}\times {BO}_{1}$, then what is the principal ${O}_{1}\times {O}_{1}$-bundle associated to $S({\gamma}_{1}\times {\gamma}_{1})\to {BO}_{1}\times {BO}_{1}$?
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$\begingroup$ Concerning your last question, the principal bundle is just $EO(1)\times EO(1)$. The principal bundle comes first; you don't need to go through the vector or sphere bundles. $\endgroup$– Alex DegtyarevCommented Dec 9, 2014 at 20:14
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The associated principal bundle is the set of pairs $(b,i)$ so that $i$ is an isomorphism of $E_b$ with $F$ as $G$-space.
answered Dec 9, 2014 at 19:37