Real rotation orbits on a complex unit quadric ($M_r\cong SO(3)(r>0),M_0\cong S^2$)

For $z=x+iy\in {\mathbb{C}}^3$ satisfying $z^{T}z=1$, the real and imaginary parts obey $|x{|}^2-|y{|}^2=1$ and $x\cdot y=0$. Simultaneous real rotations preserve $r=|y|$. At $r=0$ the group orbit is the unit sphere, with stabilizer subgroup $SO(2)$. At each $r>0$, the normalized pair $(x,y)$ is an oriented two-frame, giving a free transitive SO(3) group action. The group orbit space is $[0,\infty )$, and the whole quadric is diffeomorphic to the tangent bundle of $S^2$. Its varying Hermitian norm prevents it from being a single SU(3) group orbit.