A note on real algebraic groups

The efficacy of using complexifications to understand the structure of real algebraic groups is demonstrated. In particular, the following two results are proved: (i) Let G be a connected solvable linear group whose eigenvalues are all real. If the complexification G(C) of G is algebraic and operates algebraically on a complex variety V, and some G orbit in V is compact, then this orbit is a point. (ii) If L is a connected subgroup of a connected real linear semisimple group G such that the complexification L-C of L is algebraic and L-C contains a maximal torus of G(C), then L contains a maximal torus of G which complexifies to a maximal torus of G(C).