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 $R_0$?