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

Let A be an Artinian local ring with algebraically closed residue field k, and let be an affine smooth group scheme over A. The Greenberg functor F associates to a linear algebraic group over k, such that . We prove that if is a reductive group scheme over A, and is a maximal torus of , 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 is reductive and is a parabolic subgroup of , then P is a self-normalising subgroup of G, and if and are two Borel subgroups of , then the corresponding subgroups B and B′ are conjugate in G.