  The classification of isoparametric hypersurfaces with four principal
curvatures in spheres in [2] 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 M\"{u}nzner. The proof of the characterization in [2] 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-M\"{u}nzner polynomial. Along the way the geometric meaning of these
four sets of equations also becomes clear.
