A new look at Condition A

Ozeki and Takeuchi [14] introduced the notion of Condition A and Condition B to construct two classes of inhomogeneous isoparametric hypersurfaces with four principal curvatures in spheres, which were later generalized by Ferus, Karcher and Münzner to many more examples via the Clifford representations; we will refer to these examples of Ozeki and Takeuchi and of Ferus, Karcher and Münzner collectively as OT-FKM type throughout the paper. Dorfmeister and Neher [5] then employed isoparametric triple systems [3, 4], which are algebraic in nature, to prove that Condition A alone implies the isoparametric hypersurface is of OT-FKM type. Their proof for the case of multiplicity pairs \{3, 4\} and \{7, 8\} rests on a fairly involved algebraic classification result [9] about composition triples. In light of the classification [2] that leaves only the four exceptional multiplicity pairs \{4, 5\}, \{3, 4\}, \{7, 8\} and \{6, 9\} unsettled, it appears that Condition A may hold the key to the classification when the multiplicity pairs are \{3, 4\} and \{7, 8\}. Thus Condition A deserves to be scrutinized and understood more thoroughly from different angles. In this paper, we give a fairly short and rather straightforward proof of the result of Dorfmeister and Neher, with emphasis on the multiplicity pairs \{3, 4\} and \{7, 8\}, based on more geometric considerations. We make it explicit and apparent that the octonion algebra governs the underlying isoparametric structure.