Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 115 / 3 / c / Solution

The points $(x,y)$ are ordered orthonormal two-frames, so $P$ is the Stiefel manifold $V_2({\mathbb{R}}^{n+1})$. Changing an orthonormal basis of the same plane gives the displayed free right $O(2)$-action. Since $O(2)$ is compact, the action is proper, and the quotient is the Grassmannian of unoriented two-planes. The Free proper Lie-group action theorem makesa principal $O(2)$-bundle.