It is shown that the generalized Hopf map H x H ¿ H x R x R in quaternion formulation can be interpreted as an SO(3) orbit map for a symplectic SO(3) action. As a consequence the generalized Hopf fibration S^7 ¿ S^4 appears in the SO(3) geometric symplectic reduction of the 4DOF isotropic harmonic oscillator.
Keywords: Hopf map, Hopf fibration, symplectic reduction, harmonic oscillator
