Listing has recently extended results of Kozameh, Newman, and Tod for fourdimensional
space–times and presented a set of necessary and sufficient conditions
for a metric to be locally conformally equivalent to an Einstein metric in all semi-
Riemannian spaces of dimension nù4—subject to a nondegeneracy restriction on
the Weyl tensor. By exploiting dimensionally dependent identities we demonstrate
how to construct two alternative versions of these necessary and sufficient conditions
which we believe will be useful in applications. The four-dimensional case is
discussed in detail and examples are also given in five and six dimensions.