  In a recent paper Carot et al. considered carefully the definition of
cylindrical symmetry as a specialisation of the case of axial symmetry. One of
their propositions states that if there is a second Killing vector, which
together with the one generating the axial symmetry, forms the basis of a
two-dimensional Lie algebra, then the two Killing vectors must commute, thus
generating an Abelian group. In this comment a similar result, valid under
considerably weaker assumptions, is recalled: any two-dimensional Lie
transformation group which contains a one-dimensional subgroup whose orbits are
circles, must be Abelian. The method used to prove this result is extended to
apply to three-dimensional Lie transformation groups. It is shown that the
existence of a one-dimensional subgroup with closed orbits restricts the
Bianchi type of the associated Lie algebra to be I (Abelian), II, III, VII_0,
VIII or IX. The relationship between the present approach and that of the
original paper is discussed.
