A new look at Condition A

Ozeki and Takeuchi \cite[I]{OT} 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 \cite{DN} then employed isoparametric triple systems \cite{DN1}, 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 \cite{Mc} about composition triples. In light of the classification \cite{CCJ} 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 octonian algebra governs the underlying isoparametric structure.