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In this sequel to an earlier article, employing more commutative algebra than previously, we show that an isoparametric hypersurface with four principal curvatures and multiplicities (3,4) in S 15 is one constructed by Ozeki and Takeuchi and Ferus, Karcher, and Münzner, referred to collectively as of OT-FKM type. In fact, this new approach also gives a considerably simpler proof, both structurally and technically, that an isoparametric hypersurface with four principal curvatures in spheres withdoi:10.1215/00277630-1431813 fatcat:at5v7rytyrfaxcaitz5ryc2mhy