A comment on differential identities for the Weyl spinor perturbations

It is shown that in a type-D vacuum space-time with cosmological constant, the components of the Weyl spinor perturbations along the principal spinors of the background conformal curvature satisfy differential identities, which are valid in all the normalized spin frames {o(A),i(A)} such that o(A) and i(A) are double principal spinors of the background conformal curvature.