The classification of isoparametric hypersurfaces with four principal curvatures in spheres in  hinges on a crucial characterization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an isoparametric hypersurface of the type constructed by Ferus, Karcher and Münzner. The proof of the characterization in  is an extremely long calculation by exterior derivatives with remarkable cancellations, which is motivated by the idea that an isoparametric hypersurface is defined by an over-determined system of partial differential equations. Therefore, exterior differentiating sufficiently many times should gather us enough information for the conclusion. In spite of its elementary nature, the magnitude of the calculation and the surprisingly pleasant cancellations make it desirable to understand the underlying geometric principles.
In this paper, we give a conceptual, and considerably shorter, proof of the characterization based on Ozeki and Takeuchi's expansion formula for the Cartan-Münzner polynomial. Along the way the geometric meaning of these four sets of equations also becomes clear.
"Isoparametric hypersurfaces with four principal curvatures revisited." Nagoya Math. J. 193 129 - 154, 2009.