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