  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 $\tilde G$ which is a disjoint
union of the split Cartan subgroups associated to semisimple portions of Levi
factors of all standard parabolic subgroups of $\tilde G$. 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 ${\tilde G}/B$ with Borel subgroup $B$ of $\tilde
G$. 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.
