# Finite order automorphisms of Lie algebras

Let $\Gamma$ be a Dynkin diagram automorphism of diagram type $A_{2n}$ and let $\sigma$ be a non-trivial finite order automorphism of $\Gamma$. Let $g$ the Lie algebra associated to $\Gamma$ and consider the usual decomposition $g=g_0+g_1$. Denote the root system of $g$ by $R$ and the root system of $g_0$ by $R_0$.

How to characterize the short roots of $R_0$?

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What is the usual decomposition of $g$? –  Mariano Suárez-Alvarez Jan 2 '12 at 0:36