2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/79537The 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.49 pages, AMSTeX, Rport no: OSU MRI-99-17, corrected some typos, added some references and two figures, sbmitted to PMJ (pmj.cls is used)Symplectic GeometryExactly Solvable and Integrable Systemstext