Isoparametric hypersurfaces with four principal curvatures, III
  The classification work [5], [9] left unsettled only those anomalous isoparametric hypersurfaces with four principal curvatures and multiplicity pair $\{4,5\},\{6,9\}$ or $\{7,8\}$ in the sphere.   By systematically exploring the ideal theory in commutative algebra in conjunction with the geometry of isoparametric hypersurfaces, we show that an isoparametric hypersurface with four principal curvatures and multiplicities $\{4,5\}$ in $S^{19}$ is homogeneous, and, moreover, an isoparametric hypersurface with four principal curvatures and multiplicities $\{6,9\}$ in $S^{31}$ is either the inhomogeneous one constructed by Ferus, Karcher and M\"{u}nzner, or the one that is homogeneous.   This classification reveals the striking resemblance between these two rather different types of isoparametric hypersurfaces in the homogeneous category, even though the one with multiplicities $\{6,9\}$ is of the type constructed by Ferus, Karcher and M\"{u}nzner and the one with multiplicities $\{4,5\}$ stands alone by itself. The quaternion and the octonion algebras play a fundamental role in their geometric structures.   A unifying theme in [5]. [9] and the present sequel to them is Serre's criterion of normal varieties. Its technical side pertinent to our situation that we developed in [5], [9] and extend in this sequel is instrumental.   The classification leaves only the case of multiplicity pair $\{7,8\}$ open. 
Quo-Shin Chi
math.DG