In a recent paper, Carot et al considered carefully the definition of
cylindrical symmetry as a specialization 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, VIIq = 0, VIII or IX.
The relationship between the present approach and that of
the original paper is discussed.