The Iwasawa decomposition g = t(0) circle plus a(0) circle plus n(0) of the real semisimple Lie algebra g(0) comes from its Cartan decomposition g(0) = t(0) circle plus p(0). Then we get g(0) = t(0) circle plus b(0) where b(0) = a(0) circle plus n(0). In this note, we establish an explicite formula for the index, ind b, of b, where b is the complexification of b(0). More precisely, we show the following result:
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