Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 1 / 11E / b

By point transitivity, it suffices to take $P=i$. The conditionis equivalent to $d=a$, $c=-b$, and the determinant condition becomes $a^2+b^2=1$. Thus the stabilizer consists ofmodulo $\pm I$. Writing $a=\cos \phi$, $b=\sin \phi$ identifies it with rotations of the tangent plane, and the doubled matrix angle removes the $\pm I$ quotient. Hence