 Oriented orthonormal frame bundle of anti-de Sitter spacetime

Represent the Anti-de Sitter spacetime quadric by $x\cdot x=-{\ell }^2$ in ${\mathbb{R}}^{3,2}$. Appending its timelike unit normal $x/\ell$ to an oriented tangent pseudo-orthonormal frame identifies the frame bundle with $SO(3,2)$. The stabilizer of that normal is $SO(3,1)$, so projection realizes $SO(3,2)/SO(3,1)$ as the base of a principal bundle. For the commonly used universal cover of the anti-de Sitter quadric, pull this bundle back along the covering map; the quadric itself has a periodic timelike coordinate.