We characterize the adjoint orbits in the Lie algebra of as fibered spaces over with fibers given by the complex Cartan hypersurfaces. This combines the isoparametric hypersurfaces of case with case . The fibrations on two singular orbits turn out to be diffeomorphic to the twistor fibrations of and . From the symplectic point of view, we show that there exists a 2-parameter family of Lagrangian submanifolds on every orbit.
"Geometry of orbits and isoparametric hypersurfaces." Nagoya Math. J. 203 175 - 189, September 2011. https://doi.org/10.1215/00277630-1331899