On invariant control sets for control systems on $S^3$

In this paper we describe the Lie-theoretic structure of $\mathrm{S}\mathrm{O}(1,4)$ and consider control systems given by certain vector fields of $\mathrm{S}\mathrm{O}(1,4)$. Then we explicitly describe its invariant control sets in the unique $\mathrm{S}\mathrm{O}(1,4)$-flag manifold, namely the sphere $S^3$.