Listing has recently extended results of Kozameh, Newman, and Tod for fourdimensionalspace-times and presented a set of necessary and sufficient conditionsfor a metric to be locally conformally equivalent to an Einstein metric in all semi-Riemannian spaces of dimension nù4-subject to a nondegeneracy restriction onthe Weyl tensor. By exploiting dimensionally dependent identities we demonstratehow to construct two alternative versions of these necessary and sufficient conditionswhich we believe will be useful in applications. The four-dimensional case isdiscussed in detail and examples are also given in five and six dimensions.