Abstract: This paper studies invariant control sets for d dimensional bilinear control systems on the Stiefel manifolds St2(d) of two orthonormal frames in R?. lt is shown that under an accessibility assumption the only non trivial case which remains to be analyzed is the case where the system's semigroup is a semigroup with non void interior in Sl(d,R). For this case the quantity of invariant control sets on St 2 (d) is computed with the aid of the results of [7] about invariant control sets on the flag manifolds. Also, it is given an answer to the question of the existence or not, in an invariant control set on St2(d), of frames which span the same two dimensional subspace and are positively oriented inside this subspace. This question is suggested by the rotation numbers in higher dimensions as discussed in [2] Ver menos