The main purpose of this paper is to provide an explicit description of the invariant control sets for a class of control systems induced on the unit quaternion sphere \(S^3\) by the action of the Lorentz group \(\textrm{SO}(1,4)\) and then generalize it to the sphere \(S^{n-1}\) . These control sets are the maximal subsets of approximate controllability for the control systems. Describing them in detail is generally challenging due to the complexity of the geometry and topology of the underlying differentiable manifold and the behavior of the vector fields defining the control system. In this work, the Lie theory and the quaternions play a fundamental role in achieving our main results.
