  In this paper, we obtain a Cartan type identity for curvature-adapted
isoparametric hypersurfaces in symmetric spaces of compact type or non-compact
type. This identity is a generalization of Cartan-D'Atri's identity for
curvature-adapted(=amenable) isoparametric hypersurfaces in rank one symmetric
spaces. Furthermore, by using the Cartan type identity, we show that certain
kind of curvature-adapted isoparametric hypersurfaces in a symmetric space of
non-compact type are principal orbits of Hermann actions.
