Conjugation of semisimple subgroups over real number fields of bounded degree

Let G be a linear algebraic group over a field k of characteristic 0. We show that any two connected semisimple k-subgroups of G that are conjugate over an algebraic closure of k are actually conjugate over a finite field extension of k of degree bounded independently of the subgroups. Moreover, if k is a real number field, we show that any two connected semisimple ksubgroups of G that are conjugate over the field of real numbers R are actually conjugate over a finite real extension of k of degree bounded independently of the subgroups.