In this paper under review, by systematically exploring the ideal theory in commutative algebra in conjunction with the geometry of isoparametric hypersurfaces, the author shows that an isoparametric hypersurface with four principal curvatures and multiplicities \(\{ 4,5 \}\) in \( S^{19} \) is homogeneous, and an isoparametric hypersurface with four principal curvatures and multiplicities \( \{ 6,9 \}\) in \( S^{31} \) is either the inhomogeneous one constructed by Ferus, Karcher and Münzner, or the one that is homogeneous.
For Parts I and II by the author see [Zbl 1165.53032; Zbl 1243.53094]. See also [T. E. Cecil, the author and G. R. Jensen, Ann. Math. (2) 166, No. 1, 1–76 (2007; Zbl 1143.53058)].