One can devise an example such that there is a correspondence, but this is optional.

However, the choice of a specific O that selects observables Sx2, Sy2, Sz2 at the same time selects three orthogonal rays in physical space, namely by fixing a coordinate system ±x, ±y, ±z (which defines along which orthogonal rays the squared spin components are to be measured) in physical space. So now, by choice of an observable O, there is a direct connection of directions in space with directions in H3: orthogonality in H3 now does correspond to orthogonality in physical space. The same hol