The paper concerns the topology of an isospectral real
smooth manifold for certain Jacobi element associated with real split semisimple Lie
algebra. The manifold is identified as a compact, connected completion of the
disconnected Cartan subgroup of the corresponding Lie group G which is a disjoint
union of the split Cartan subgroups associated to semisimple portions of
Levi factors of all standard parabolic subgroups of G. The manifold is also
related to the compactified level sets of a generalized Toda lattice equation
defined on the semisimple Lie algebra, which is diffeomorphic to a toric
variety in the flag manifold G∕B with Borel subgroup B of G. We then give a
cellular decomposition and the associated chain complex of the manifold by
introducing colored-signed Dynkin diagrams which parametrize the cells in the
decomposition.
