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The generalized Borel-de Siebenthal theorem

Abstract Let D be a Dynkin diagram. The classical Borel-de Siebenthal theorem uses an algorithm to simplify every painting on D to one with a single black vertex. In this article, we generalize this theorem to elements of Zn, so that n=2 corresponds to the paintings. We also explain the algebraic significance of this result in terms of automorphisms on complex simple Lie algebras.