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 $S({\gamma}\_{1}\times {\gamma}\_{1})\to {BO}\_{1}\times {BO}\_{1}$?



Principal bundle associated to a fiber bundle
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.


