Levi Factors and Admissible Automorphisms

Let g be a complex simple Lie algebra. We consider subalgebras m which are Levi factors of parabolic subalgebras of g, or equivalently m is the centralizer of its center. We introduced the notion of admissible systems on finite order g-automorphisms θ , and show that θ has admissible systems if and only if its fixed point set is a Levi factor. We then use the extended Dynkin diagrams to characterize