GL(V+) is a C-isomorphism, where GL(V, I) is viewed as a C-group. Now, U is a (Zariski connected) reductive R-subgroup of GL(V, I) (viewed as an R-group in GL(V)=GL,n(R)), so that Uc is a (Zariski connected) reductive C-subgroup ofGL(Vc, I) (in GL(Vc)=GL,n(C)). First we want to show that the restriction map rr induces a C-isomorphism of Uc onto its image in GL(V+). It suffices to show that rr:\Uc is injective. If X, Yeu, one has X+iY\V+=O=>X+IY =0=>X = Y =0, because u is a real form of g.
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