A quick proof on the equivalence classes of extended Vogan diagrams

An extended Vogan diagram is an extended Dynkin diagram together with a diagram involution, such that the vertices fixed by the involution are colored white or black. Every extended Vogan diagram represents an almost compact real form of the affine Kac-Moody Lie algebra. Two extended diagrams are said to be equivalent if they represent isomorphic real forms. The equivalence classes of extended Vogan diagrams have earlier been classified by the authors. In this paper, we present a much shorter and instructive argument.