Cartan Decompositions and Semigroups of Simple Lie Groups

Let G be a split real connected simple Lie group and S a semigroup of G that contains a subgroup G() for an arbitrary root , isomorphic to SL(2, R) .We present a Cartan decomposition of the Lie algebra of G, related to , invariant by the adjoint action of the Lie algebra sl(2, R) that allows to characterize some properties of the Lie saturate of the semigroup S .We give necessary and sufficient conditions for S to be equal to the whole group G.