  For Petrov D vacuum spaces, two simple identities are rederived and some new
identities are obtained, in a manageable form, by a systematic and transparent
analysis using the GHP formalism. This gives a complete involutive set of
tables for the four GHP derivatives on each of the four GHP spin coefficients
and the one Weyl tensor component. It follows directly from these results that
the theoretical upper bound on the order of covariant differentiation of the
Riemann tensor required for a Karlhede classification of these spaces is
reduced to two.

General Relativity and Quantum Cosmology