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124 9 Invariants and the characterization of geometries9.3.2 The remainingstepsHaving thus dealt with step 2 for q = 0 of the procedure given in theprevious section, we now have to bring the Riemann tensor into canonicalform by Lorentz transformations. Although this is in principle algorithmic(since quartics have an algorithmic solution, and the quartic forPetrov classification has as its roots the parameters of the required nullrotations), it is in practice often the most difficult step, especially in thePetrov type I case. In algebraically special cases, we are helped by the factthat the method used for Petrov classification can be related to theEuclidean algorithm for simultaneous roots of polynomials, so that whenthere are equal roots they are readily found.The fourth and fifth steps are straightforward in principle. Functionaldependence is tested by finding the rank of the Jacobians between thepossibly independent functions, and the isotropy can be found from thecanonical form.Now we reach step 1 again for q =1.Forq ≥ 1 it is very importantin calculations to cut down the number of quantities to be computed bytaking only a minimal set of derivatives of the Riemann tensor, which canbe done because the derivatives obey the Ricci and Bianchi identities. Anexplicit minimal set of derivatives was found by MacCallum and Åman(1986), generalizing the treatment of the electrovac case in Penrose (1960).In terms of the Newman–Penrose quantities Ψ ABCD , Φ AB ȦḂ and Λ it canbe described as follows. One has to take, in the general case, the followingcomponents for q ≥ 0 (where the term ‘totally symmetrized’ means thata spinor is to be symmetrized over all its free dotted indices and over allits free undotted indices):1. The totally symmetrized qth derivatives of Λ.2. The totally symmetrized qth derivatives of Ψ ABCD .3. The totally symmetrized qth derivatives of Φ AB4. For q ≥ 1, the totally symmetrized (q − 1)th derivatives ȦḂ.ofΞ ABC Ḋ = ∇D Ḋ Ψ DABC which is one side of one of the Bianchiidentities.5. For q ≥ 2, the d’Alembertian ∇ AA˙∇ A A˙applied to all the quantitiescalculated for the derivatives of order q − 2.Steps 2 and 3 of the procedure for q ≥ 1 are handled in existing programsin a manner which is capable of refinements that would probablybe useful in creating more precise classifications. In fact, in most cases allthat is tested is whether any invariance at the q = 0 stage persists, and ifit does not, new canonical forms are not found. Steps 4 and 5 continue tobe in principle straightforward, if more time-consuming (because of theincreasing numbers of components to be tested).