Isoparametric hypersurfaces with four principal curvatures, II

In this sequel, employing more commutative algebra than that explored in \cite{CCJ}, we show that an isoparametric hypersurface with four principal curvatures and multiplicities $(3,4)$ in $S^{15}$ is one constructed by Ozeki-Takeuchi \cite[I]{OT} and Ferus-Karcher-M\"unzner \cite{FKM}, referred to collectively as of OT-FKM type. In fact, this new approach also gives a considerably simpler, both structurally and technically, proof \cite{CCJ} that an isoparametric hypersurface with four principal curvatures in spheres with the multiplicity constraint $m_2\geq 2m_1-1$ is of OT-FKM type, which left unsettled exactly the four anomalous multiplicity pairs $(4,5),(3,4),(7,8)$ and $(6,9)$, where the last three are closely tied, respectively, with the quaternion algebra, the octonion algebra and the complexified octonion algebra, whereas the first stands alone by itself in that it cannot be of OT-FKM type. A byproduct of this new approach is that we see that Condition B, introduced by Ozeki and Takeuchi \cite[I]{OT} in their construction of inhomogeneous isoparametric hypersurfaces, naturally arises. The cases for the multiplicity pairs $(4,5),(6,9)$ and $(7,8)$ remain open now.