Bianchi and Weyl equations as sufficient equations for Einstein spaces (Q1842090)

The author assumes the Bianchi identities and the Weyl wave equations (in the form which they take for an Einstein space) for an arbitrary symmetric 4-spinor, with the background space being unspecified. It is then proven that -- in general -- the consistency of these equations demands that the background space be an Einstein space and that the arbitrary symmetric 4-spinor be the Weyl 4-spinor. In other words, the Bianchi identities and the Weyl wave equations are sufficient conditions for an Einstein space. The implications of this result for the Newman- Penrose and Geroch-Held-Penrose spin-coefficient formalisms are briefly discussed.