A comment on a paper by Carot et al
  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. 
Alan Barnes (Aston University)