Reductive group schemes, the Greenberg functor, and associated algebraic groups

AbstractLet A be an Artinian local ring with algebraically closed residue field k, and let G be an affine smooth group scheme over A. The Greenberg functor F associates to G a linear algebraic group G≔(FG)(k) over k, such that G≅G(A). We prove that if G is a reductive group scheme over A, and T is a maximal torus of G, then T is a Cartan subgroup of G, and every Cartan subgroup of G is obtained uniquely in this way. Moreover, we prove that if G is reductive and P is a parabolic subgroup of G, then P is a self-normalising subgroup of G, and if B and B′ are two Borel subgroups of G, then the corresponding subgroups B and B′ are conjugate in G