Petrov D vacuum spaces revisited: Identities and Invariant   Classification
  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. 
S. Brian Edgar, Alfonso Garc\'ia-Parrado G\'omez-Lobo and Jos\'e M.   Mart\'in-Garc\'ia