Following the lines of his paper reviewed above, the author gives various conditions that are necessary or sufficient (sometimes both) for the controllability of invariant affine control systems on Lie groups. The main hypothesis made is the existence of codimension-one subgroups what makes the author's ``hypersurface principle'' work. For an earlier exploitation of this principle, the reader may want to look at the author's paper reviewed above [ibid. 2, 55-67 (1996; Zbl 0991.93014)], \textit{J. Hilgert} and \textit{K. Hofmann}'s paper ``On the causal structure of homogeneous manifolds'' [ Math. Scand. 67, 119-144 (1990; Zbl 0739.53041)], or \textit{J. Hilgert}'s paper ``The halfspace method for causal structures on homogeneous manifolds'' [in: K. H. Hofmann et al. (eds.), Semigroups in algebra, geometry and analysis, de Gruyter Expo. Math. 20, 33-55 (1995; Zbl 0848.22006)].