  In the study of induced bilinear systems, the classical Lie algebra rank
condition (LARC) is known to be impractical since it requires computing the
rank everywhere. On the other hand, the transitive Lie algebra condition, while
more commonly used, relies on the classification of transitive Lie algebras,
which is elusive except for few simple geometric objects such as spheres. We
prove in this note that for bilinear systems induced by proper Lie group
actions, the underlying Lie algebra is closely related to the orbits of the
group action. Knowing the pattern of the Lie algebra rank over the manifold, we
show that the LARC can be relaxed so that it suffices to check the rank at an
arbitrary single point. Moreover, it removes the necessity for classifying
transitive Lie algebras. Finally, this relaxed rank condition also leads to a
characterization of controllable submanifolds by orbits.
