  We study metric transformations including not just the field strength tensor
of a $U(1)$ gauge field, but also its dual tensor. We first consider an
arbitrary symmetric matrix built up with these two tensors in the metric
transformation. It turns out the form of transformation reduces to a quite
simple form on imposing the parity evenness of the transformed metric and by
utilising the Cayley-Hamilton theorem as well as other useful identities.
Interestingly, the same form for the transformation was recently argued in the
process of seeking for a generic metric transformation but without the
inclusion of the dual tensor.
