The lattice of characteristic subspaces of an endomorphism with Jordan-Chevalley decomposition

Given an endomorphism A over a finite dimensional vector space having Jordan-Chevalley decomposition, the lattices of invariant and hyperinvariant subspaces of A can be obtained from the nilpotent part of this decomposition. We extend this result for lattices of characteristic subspaces. We also obtain a generalization of Shoda's theorem about the characterization of the existence of characteristic non hyperinvariant subspaces.