Folding by Automorphisms

Let Φ be a simply-laced root system with simple roots ∆ = {αi : i ∈ I} embedded in the real vector space V with inner product 〈·,·〉. For convenience, we may assume that the roots have been normalized so that 〈α, α〉 = 2 for all α ∈ Φ. As a second convenience, we may assume that V has been enlarged, if necessary, so that 〈·,·〉 is nondegenerate on V . Now let σ be a diagram automorphism of (Φ,∆). Given our choice of normalization, this amounts to a permutation of I such that