Calling Sequences

Parameters

W         - a symmetric rank 4 covariant spinor

PT        - the Petrov type of the spinor $W$

options   - one or more of the keyword arguments method and output

Description

 • Let ${W}_{\mathrm{ABCD}}$ be a rank 4 symmetric spinor and let  be a solder form on a 4-dimensional spacetime of signature [1, -1, -1, -1]. Then the rank 4 tensor

=

enjoys all the symmetries of the Weyl tensor. It is skew-symmetric in the indices $\mathrm{ab}$ and satisfies the cyclic identity on $\mathrm{bcd},$ and is trace-free with respect to the metric defined by the solder form $\mathrm{σ}$.

 • If is a spinor dyad (a pair of rank-2 spinors with ) then the spinor can be expressed as

.    (*)

The coefficients coincide with the Newman-Penrose Weyl scalars for the Weyl tensor constructed from the null tetrad defined by the spinor dyad

 • The Petrov type of the spinor ${W}_{\mathrm{ABCD}}$ coincides with the Petrov type of the Weyl tensor or the Petrov type of the spacetime determined by the Newman-Penrose Weyl scalars. See NPCurvatureScalars, NullTetrad, PetrovType, SolderForm, WeylSpinor.
 • The normal forms for the Newman-Penrose coefficients are as follows.

Type I.

Type II. ${\mathrm{Ψ}}_{0}$

Type III.

Type D.

Type N.

Type O.

See Penrose and Rindle Vol. 2, Section 8.3.

Thus, for example, if the Petrov type of the Weyl spinor is $\mathrm{D}$, then there is a spinor dyad such that the Weyl spinor takes the form . Such a spinor dyad is said to be an adapted spinor dyad.

 • The command AdaptedSpinorDyad returns a contravariant spinor dyad which will put the given Weyl spinor in the above normal form. If the Petrov type is I, then the values of are also returned. If the Petrov type is II or D, the value of is given.
 • The adapted spinor dyads need not be unique; multiple adapted spinor dyads may be returned. All computed dyads will be returned with the keyword argument output = $"all"$. If the type is III or N, then a quasi-adapted dyad (where the 1 non-zero Newman-Penrose coefficient is not normalized to 1) will be calculated if method = "unnormalized".
 • The command AdaptedSpinorDyad is part of the DifferentialGeometry:-Tensor package. It can be used in the form AdaptedSpinorDyad(...) only after executing the commands with(DifferentialGeometry) and with(Tensor), but can always be used by executing DifferentialGeometry:-Tensor:-AdaptedSpinorDyad(...).

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{Tensor}\right):$

Set the global environment variable _EnvExplicit to true to insure that our results are free of expressions.

 Spin > $\mathrm{_EnvExplicit}≔\mathrm{true}:$

We give examples of Weyl spinors of each Petrov type and calculate an adapted spinor dyad. We check that the spinor dyad has the desired properties.

First create the spinor bundle over a 4 dimensional spacetime.

 > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],\left[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right],\mathrm{Spin}\right)$
 ${\mathrm{frame name: Spin}}$ (2.1)

In order to construct the Weyl spinors for our examples, we need a basis for the vector space of symmetric rank 4 spinors. This we obtain from the GenerateSymmetricTensors command.

 Spin > $S≔\mathrm{GenerateSymmetricTensors}\left(\left[\mathrm{dz1},\mathrm{dz2}\right],4\right)$
 ${S}{:=}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{6}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{5}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{5}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{5}\right]{,}\frac{{1}}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{6}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{5}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{5}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{5}\right]{,}\frac{{1}}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{6}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{5}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{5}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{5}\right]{,}\frac{{1}}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{6}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{5}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{5}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{5}\right]{,}\frac{{1}}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{6}{,}{6}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{5}\right]{,}\frac{{1}}{{6}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{6}{,}{6}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{5}\right]{,}\frac{{1}}{{6}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{6}{,}{6}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{5}\right]{,}\frac{{1}}{{6}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{6}{,}{6}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{5}\right]{,}\frac{{1}}{{6}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{6}{,}{6}{,}{6}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{6}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{6}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{5}\right]{,}\frac{{1}}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{6}{,}{6}{,}{6}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{6}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{6}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{5}\right]{,}\frac{{1}}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{6}{,}{6}{,}{6}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{6}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{6}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{5}\right]{,}\frac{{1}}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{6}{,}{6}{,}{6}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{6}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{6}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{5}\right]{,}\frac{{1}}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.2)

Example 1. Type I

Define a rank 4 spinor ${W}_{1}.$

 Spin > $\mathrm{W1}≔\mathrm{DGzip}\left(\left[6,12,30,24,6\right],S,"plus"\right)$
 ${\mathrm{W1}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{6}\right]{,}\left[\left[{5}{,}{5}{,}{5}{,}{6}\right]{,}{3}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{5}\right]{,}{3}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{6}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{5}\right]{,}{3}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{5}\right]{,}{3}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}{5}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{5}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{5}\right]{,}{5}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{5}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{6}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{6}\right]{,}\left[\left[{5}{,}{5}{,}{5}{,}{6}\right]{,}{3}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{5}\right]{,}{3}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{6}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{5}\right]{,}{3}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{5}\right]{,}{3}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}{5}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{5}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{5}\right]{,}{5}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{5}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{6}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{6}\right]{,}\left[\left[{5}{,}{5}{,}{5}{,}{6}\right]{,}{3}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{5}\right]{,}{3}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{6}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{5}\right]{,}{3}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{5}\right]{,}{3}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}{5}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{5}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{5}\right]{,}{5}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{5}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{6}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{6}\right]{,}\left[\left[{5}{,}{5}{,}{5}{,}{6}\right]{,}{3}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{5}\right]{,}{3}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{6}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{5}\right]{,}{3}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{5}\right]{,}{3}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}{5}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{5}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{5}\right]{,}{5}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{5}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{6}\right]\right]\right]\right)$ (2.3)

Calculate the Newman-Penrose coefficients for ${W}_{1}$ with respect to the initial dyad basis .

 Spin > $\mathrm{NP1}≔\mathrm{NPCurvatureScalars}\left(\mathrm{W1},\left[\mathrm{dz1},\mathrm{dz2}\right]\right)$
 ${\mathrm{NP1}}{:=}{\mathrm{table}}\left(\left[{"Psi2"}{=}{5}{,}{"Psi0"}{=}{6}{,}{"Psi1"}{=}{-}{6}{,}{"Psi3"}{=}{-}{3}{,}{"Psi4"}{=}{6}\right]\right)$ (2.4)

Use these coefficients to find the Petrov type of ${W}_{1}.$

 Spin > $\mathrm{PetrovType}\left(\mathrm{NP1}\right)$
 ${"I"}$ (2.5)

 Spin > $\mathrm{SpinorDyadCon1},\mathrm{η1},\mathrm{χ1}≔\mathrm{AdaptedSpinorDyad}\left(\mathrm{W1},"I"\right)$
 ${\mathrm{SpinorDyadCon1}}{,}{\mathrm{η1}}{,}{\mathrm{χ1}}{:=}\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]{,}\left[\left[{6}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]{,}\left[\left[{6}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]{,}\left[\left[{6}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]{,}\left[\left[{6}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{6}\right]{,}{1}\right]\right]\right]\right)\right]{,}{1}{,}{4}$ (2.6)

Here is the covariant form of the spinor dyad.

 Spin > $\mathrm{Dyad1}≔\mathrm{map}\left(\mathrm{RaiseLowerSpinorIndices},\mathrm{SpinorDyadCon1},\left[1\right]\right)$
 ${\mathrm{Dyad1}}{:=}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]{,}\left[\left[{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]{,}\left[\left[{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]{,}\left[\left[{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]{,}\left[\left[{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}\right]{,}{-1}\right]\right]\right]\right)\right]$ (2.7)

We check that this answer is correct in two ways. First we can re-calculate the Newman-Penrose coefficients and confirm that they are in the correct normal form.

 Spin > $\mathrm{NPCurvatureScalars}\left(\mathrm{W1},\mathrm{Dyad1}\right)$
 ${\mathrm{table}}\left(\left[{"Psi2"}{=}{-}{1}{,}{"Psi0"}{=}{6}{,}{"Psi1"}{=}{0}{,}{"Psi3"}{=}{0}{,}{"Psi4"}{=}{6}\right]\right)$ (2.8)

This is the correct normal form since and

Second, we can calculate a type I Weyl spinor from this spinor dyad using the command WeylSpinor and check that the result coincides with the original spinor ${W}_{1}$.

 Spin > $\mathrm{W1Check}≔\mathrm{WeylSpinor}\left(\mathrm{Dyad1},"I",1,4\right)$
 ${\mathrm{W1Check}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{6}\right]{,}\left[\left[{5}{,}{5}{,}{5}{,}{6}\right]{,}{3}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{5}\right]{,}{3}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{6}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{5}\right]{,}{3}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{5}\right]{,}{3}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}{5}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{5}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{5}\right]{,}{5}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{5}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{6}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{6}\right]{,}\left[\left[{5}{,}{5}{,}{5}{,}{6}\right]{,}{3}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{5}\right]{,}{3}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{6}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{5}\right]{,}{3}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{5}\right]{,}{3}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}{5}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{5}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{5}\right]{,}{5}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{5}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{6}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{6}\right]{,}\left[\left[{5}{,}{5}{,}{5}{,}{6}\right]{,}{3}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{5}\right]{,}{3}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{6}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{5}\right]{,}{3}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{5}\right]{,}{3}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}{5}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{5}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{5}\right]{,}{5}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{5}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{6}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{6}\right]{,}\left[\left[{5}{,}{5}{,}{5}{,}{6}\right]{,}{3}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{5}\right]{,}{3}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{6}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{5}\right]{,}{3}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{5}\right]{,}{3}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}{5}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{5}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{5}\right]{,}{5}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{5}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{6}\right]\right]\right]\right)$ (2.9)
 Spin > $\mathrm{W1}&minus\mathrm{W1Check}$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{0}\right]\right]\right]\right)$ (2.10)

Alternative dyads will be computed with the keyword argument output

 Spin > $\mathrm{_EnvExplicit}≔\mathrm{true}:$
 Spin > $\mathrm{AdaptedSpinorDyad}\left(\mathrm{W1},"I",\mathrm{output}="all"\right)$
 $\left[\left[\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{6}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{6}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{6}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{6}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]{,}\left[\left[{6}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]{,}\left[\left[{6}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]{,}\left[\left[{6}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]{,}\left[\left[{6}\right]{,}{-1}\right]\right]\right]\right)\right]{,}{1}{,}{4}\right]{,}\left[\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}\frac{{1}}{{2}}{+}\frac{{I}}{{2}}\right]{,}\left[\left[{6}\right]{,}{-1}{-}{I}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}\frac{{1}}{{2}}{+}\frac{{I}}{{2}}\right]{,}\left[\left[{6}\right]{,}{-1}{-}{I}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}\frac{{1}}{{2}}{+}\frac{{I}}{{2}}\right]{,}\left[\left[{6}\right]{,}{-1}{-}{I}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}\frac{{1}}{{2}}{+}\frac{{I}}{{2}}\right]{,}\left[\left[{6}\right]{,}{-1}{-}{I}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right]\right]\right]\right)\right]{,}{3}{,}{-}\frac{{1}}{{3}}\right]{,}\left[\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{-}\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right]{,}\left[\left[{6}\right]{,}{I}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{-}\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right]{,}\left[\left[{6}\right]{,}{I}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{-}\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right]{,}\left[\left[{6}\right]{,}{I}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{-}\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right]{,}\left[\left[{6}\right]{,}{I}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}\frac{{1}}{{2}}{+}\frac{{I}}{{2}}\right]{,}\left[\left[{6}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}\frac{{1}}{{2}}{+}\frac{{I}}{{2}}\right]{,}\left[\left[{6}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}\frac{{1}}{{2}}{+}\frac{{I}}{{2}}\right]{,}\left[\left[{6}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}\frac{{1}}{{2}}{+}\frac{{I}}{{2}}\right]{,}\left[\left[{6}\right]{,}{-1}\right]\right]\right]\right)\right]{,}{-}{4}{,}\frac{{3}}{{4}}\right]{,}\left[\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{6}\right]{,}{-I}{}\sqrt{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{6}\right]{,}{-I}{}\sqrt{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{6}\right]{,}{-I}{}\sqrt{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{6}\right]{,}{-I}{}\sqrt{{2}}\right]\right]\right]\right)\right]{,}{4}{,}\frac{{1}}{{4}}\right]{,}\left[\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{-}\frac{\sqrt{{2}}}{{2}}{+}\frac{{I}{}\sqrt{{2}}}{{2}}\right]{,}\left[\left[{6}\right]{,}\frac{\sqrt{{2}}}{{2}}{-}\frac{{I}{}\sqrt{{2}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{-}\frac{\sqrt{{2}}}{{2}}{+}\frac{{I}{}\sqrt{{2}}}{{2}}\right]{,}\left[\left[{6}\right]{,}\frac{\sqrt{{2}}}{{2}}{-}\frac{{I}{}\sqrt{{2}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{-}\frac{\sqrt{{2}}}{{2}}{+}\frac{{I}{}\sqrt{{2}}}{{2}}\right]{,}\left[\left[{6}\right]{,}\frac{\sqrt{{2}}}{{2}}{-}\frac{{I}{}\sqrt{{2}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{-}\frac{\sqrt{{2}}}{{2}}{+}\frac{{I}{}\sqrt{{2}}}{{2}}\right]{,}\left[\left[{6}\right]{,}\frac{\sqrt{{2}}}{{2}}{-}\frac{{I}{}\sqrt{{2}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{6}\right]{,}\left({-}\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right){}\sqrt{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{6}\right]{,}\left({-}\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right){}\sqrt{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{6}\right]{,}\left({-}\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right){}\sqrt{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{6}\right]{,}\left({-}\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right){}\sqrt{{2}}\right]\right]\right]\right)\right]{,}{-}{3}{,}\frac{{4}}{{3}}\right]{,}\left[\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{-}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{6}\right]{,}\frac{\sqrt{{2}}}{{2}}{-}\frac{{I}{}\sqrt{{2}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{-}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{6}\right]{,}\frac{\sqrt{{2}}}{{2}}{-}\frac{{I}{}\sqrt{{2}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{-}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{6}\right]{,}\frac{\sqrt{{2}}}{{2}}{-}\frac{{I}{}\sqrt{{2}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{-}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{6}\right]{,}\frac{\sqrt{{2}}}{{2}}{-}\frac{{I}{}\sqrt{{2}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{-}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{6}\right]{,}{-}\frac{\sqrt{{2}}}{{2}}{+}\frac{{I}{}\sqrt{{2}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{-}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{6}\right]{,}{-}\frac{\sqrt{{2}}}{{2}}{+}\frac{{I}{}\sqrt{{2}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{-}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{6}\right]{,}{-}\frac{\sqrt{{2}}}{{2}}{+}\frac{{I}{}\sqrt{{2}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{-}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{6}\right]{,}{-}\frac{\sqrt{{2}}}{{2}}{+}\frac{{I}{}\sqrt{{2}}}{{2}}\right]\right]\right]\right)\right]{,}{-}{1}{,}{-}{3}\right]\right]$ (2.11)

Example 2. Type II

Define a rank 4 spinor ${W}_{2}.$

 Spin > $\mathrm{W2}≔\mathrm{DGzip}\left(\left[4,4,6,16,10\right],S,"plus"\right)$
 ${\mathrm{W2}}{:=}{4}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{4}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{4}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{4}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{4}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{10}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.12)

Calculate the Newman-Penrose coefficients for ${W}_{2}$ with respect to the standard dyad basis .

 Spin > $\mathrm{NP2}≔\mathrm{NPCurvatureScalars}\left(\mathrm{W2},\left[\mathrm{dz1},\mathrm{dz2}\right]\right)$
 ${\mathrm{NP2}}{:=}{\mathrm{table}}\left(\left[{"Psi0"}{=}{10}{,}{"Psi1"}{=}{-}{4}{,}{"Psi3"}{=}{-}{1}{,}{"Psi4"}{=}{4}{,}{"Psi2"}{=}{1}\right]\right)$ (2.13)

Find the Petrov type of ${W}_{2}.$

 Spin > $\mathrm{PetrovType}\left(\mathrm{NP2}\right)$
 ${"II"}$ (2.14)

Compute an adapted contravariant spinor dyad for ${W}_{2}$.

 Spin > $\mathrm{SpinorDyadCon2},\mathrm{η2}≔\mathrm{AdaptedSpinorDyad}\left(\mathrm{W2},"II"\right)$
 ${\mathrm{SpinorDyadCon2}}{,}{\mathrm{η2}}{:=}\left[\frac{{1}}{{3}}{}\sqrt{{3}}{}{\mathrm{D_z1}}{-}\frac{{1}}{{3}}{}\sqrt{{3}}{}{\mathrm{D_z2}}{,}{-}\frac{{2}}{{3}}{}\sqrt{{3}}{}{\mathrm{D_z1}}{-}\frac{{1}}{{3}}{}\sqrt{{3}}{}{\mathrm{D_z2}}\right]{,}{3}$ (2.15)

 Spin > $\mathrm{Dyad2}≔\mathrm{map}\left(\mathrm{RaiseLowerSpinorIndices},\mathrm{SpinorDyadCon2},\left[1\right]\right)$
 ${\mathrm{Dyad2}}{:=}\left[\frac{{1}}{{3}}{}\sqrt{{3}}{}{\mathrm{dz1}}{+}\frac{{1}}{{3}}{}\sqrt{{3}}{}{\mathrm{dz2}}{,}\frac{{1}}{{3}}{}\sqrt{{3}}{}{\mathrm{dz1}}{-}\frac{{2}}{{3}}{}\sqrt{{3}}{}{\mathrm{dz2}}\right]$ (2.16)

We check that this answer is correct in two ways. First we can re-calculate the Newman-Penrose coefficients and confirm that they are in the correct normal form.

 Spin > $\mathrm{NPCurvatureScalars}\left(\mathrm{W2},\mathrm{Dyad2}\right)$
 ${\mathrm{table}}\left(\left[{"Psi0"}{=}{0}{,}{"Psi1"}{=}{0}{,}{"Psi3"}{=}{0}{,}{"Psi4"}{=}{18}{,}{"Psi2"}{=}{3}\right]\right)$ (2.17)

This is the correct normal form since and

Second, we can calculate a type II Weyl spinor from this spinor dyad and check that the result coincides with the original spinor ${W}_{2}$.

 Spin > $\mathrm{W2Check}≔\mathrm{WeylSpinor}\left(\mathrm{Dyad2},"II",3\right)$
 ${\mathrm{W2Check}}{:=}{4}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{4}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{4}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{4}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{4}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{10}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.18)
 Spin > $\mathrm{W2}&minus\mathrm{W2Check}$
 ${0}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}$ (2.19)

Example 3. Type III

Define a rank 4 spinor ${W}_{3}.$

 Spin > $\mathrm{W3}≔\mathrm{DGzip}\left(\left[-8,-20I,12,-4I,4\right],S,"plus"\right)$
 ${\mathrm{W3}}{:=}{-}{8}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{I}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{-}{5}{}{I}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{2}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{-}{I}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{2}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{2}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{-}{I}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{-}{5}{}{I}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{2}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{2}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{-}{5}{}{I}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{2}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{I}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{-}{5}{}{I}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{4}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.20)

Calculate the Newman-Penrose coefficients for ${W}_{3}$ with respect to the initial dyad basis .

 Spin > $\mathrm{NP3}≔\mathrm{NPCurvatureScalars}\left(\mathrm{W3},\left[\mathrm{dz1},\mathrm{dz2}\right]\right)$
 ${\mathrm{NP3}}{:=}{\mathrm{table}}\left(\left[{"Psi0"}{=}{4}{,}{"Psi1"}{=}{I}{,}{"Psi3"}{=}{5}{}{I}{,}{"Psi4"}{=}{-}{8}{,}{"Psi2"}{=}{2}\right]\right)$ (2.21)

Find the Petrov type of ${W}_{3}.$

 Spin > $\mathrm{PetrovType}\left(\mathrm{NP3}\right)$
 ${"III"}$ (2.22)

Compute an adapted contravariant spinor dyad for ${W}_{3}.$

 Spin > $\mathrm{SpinorDyadCon3}≔\mathrm{AdaptedSpinorDyad}\left(\mathrm{W3},"III"\right)$
 ${\mathrm{SpinorDyadCon3}}{:=}\left[\left({-}\frac{{1}}{{2}}{-}\frac{{1}}{{2}}{}{I}\right){}\sqrt{{6}}{}{\mathrm{D_z1}}{+}\left(\frac{{1}}{{2}}{}\sqrt{{6}}{-}\frac{{1}}{{2}}{}{I}{}\sqrt{{6}}\right){}{\mathrm{D_z2}}{,}\left({-}\frac{{1}}{{18}}{-}\frac{{1}}{{18}}{}{I}\right){}\sqrt{{6}}{}{\mathrm{D_z1}}{+}\left({-}\frac{{1}}{{9}}{+}\frac{{1}}{{9}}{}{I}\right){}\sqrt{{6}}{}{\mathrm{D_z2}}\right]$ (2.23)

Here is the covariant form of the spinor dyad.

 Spin > $\mathrm{Dyad3}≔\mathrm{map}\left(\mathrm{RaiseLowerSpinorIndices},\mathrm{SpinorDyadCon3},\left[1\right]\right)$
 ${\mathrm{Dyad3}}{:=}\left[\left({-}\frac{{1}}{{2}}{}\sqrt{{6}}{+}\frac{{1}}{{2}}{}{I}{}\sqrt{{6}}\right){}{\mathrm{dz1}}{+}\left({-}\frac{{1}}{{2}}{-}\frac{{1}}{{2}}{}{I}\right){}\sqrt{{6}}{}{\mathrm{dz2}}{,}\left(\frac{{1}}{{9}}{-}\frac{{1}}{{9}}{}{I}\right){}\sqrt{{6}}{}{\mathrm{dz1}}{+}\left({-}\frac{{1}}{{18}}{-}\frac{{1}}{{18}}{}{I}\right){}\sqrt{{6}}{}{\mathrm{dz2}}\right]$ (2.24)

We check that this answer is correct in two ways. First we can re-calculate the Newman-Penrose coefficients and confirm that they are in the correct normal form.

 Spin > $\mathrm{NPCurvatureScalars}\left(\mathrm{W3},\mathrm{Dyad3}\right)$
 ${\mathrm{table}}\left(\left[{"Psi0"}{=}{0}{,}{"Psi1"}{=}{0}{,}{"Psi3"}{=}{1}{,}{"Psi4"}{=}{0}{,}{"Psi2"}{=}{0}\right]\right)$ (2.25)

This is the correct normal form since $1.$

Second, we can calculate a type III Weyl spinor from this spinor dyad and check that the result coincides with the original spinor ${W}_{3}$.

 Spin > $\mathrm{W3Check}≔\mathrm{WeylSpinor}\left(\mathrm{Dyad3},"III"\right)$
 ${\mathrm{W3Check}}{:=}{-}{8}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{I}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{-}{5}{}{I}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{2}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{-}{I}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{2}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{2}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{-}{I}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{-}{5}{}{I}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{2}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{2}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{-}{5}{}{I}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{2}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{5}{}{I}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{I}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{4}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.26)
 Spin > $\mathrm{W3}&minus\mathrm{W3Check}$
 ${0}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}$ (2.27)

Example 4. Type D

Define a rank 4 spinor ${W}_{4}.$

 Spin > $\mathrm{W4}≔\mathrm{DGzip}\left(\left[3,-18,3,72,48\right],S,"plus"\right)$
 ${\mathrm{W4}}{:=}{3}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}\frac{{9}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{-}\frac{{9}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}\frac{{1}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{-}\frac{{9}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}\frac{{1}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}\frac{{1}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{18}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{-}\frac{{9}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}\frac{{1}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}\frac{{1}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{18}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}\frac{{1}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{18}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{18}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{48}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.28)

Calculate the Newman-Penrose coefficients for ${W}_{4}$ with respect to the initial dyad basis .

 Spin > $\mathrm{NP4}≔\mathrm{NPCurvatureScalars}\left(\mathrm{W4},\left[\mathrm{dz1},\mathrm{dz2}\right]\right)$
 ${\mathrm{NP4}}{:=}{\mathrm{table}}\left(\left[{"Psi0"}{=}{48}{,}{"Psi1"}{=}{-}{18}{,}{"Psi3"}{=}\frac{{9}}{{2}}{,}{"Psi4"}{=}{3}{,}{"Psi2"}{=}\frac{{1}}{{2}}\right]\right)$ (2.29)

Find the Petrov type of ${W}_{4}.$

 Spin > $\mathrm{PetrovType}\left(\mathrm{NP4}\right)$
 ${"D"}$ (2.30)

Compute an adapted contravariant spinor dyad for ${W}_{4}.$

 Spin > $\mathrm{SpinorDyadCon4},\mathrm{η4}≔\mathrm{AdaptedSpinorDyad}\left(\mathrm{W4},"D"\right)$
 ${\mathrm{SpinorDyadCon4}}{,}{\mathrm{η4}}{:=}\left[{\mathrm{D_z1}}{-}{\mathrm{D_z2}}{,}\frac{{4}}{{5}}{}{\mathrm{D_z1}}{+}\frac{{1}}{{5}}{}{\mathrm{D_z2}}\right]{,}\frac{{25}}{{2}}$ (2.31)

Here is the covariant form of the spinor dyad.

 Spin > $\mathrm{Dyad4}≔\mathrm{map}\left(\mathrm{RaiseLowerSpinorIndices},\mathrm{SpinorDyadCon4},\left[1\right]\right)$
 ${\mathrm{Dyad4}}{:=}\left[{\mathrm{dz1}}{+}{\mathrm{dz2}}{,}{-}\frac{{1}}{{5}}{}{\mathrm{dz1}}{+}\frac{{4}}{{5}}{}{\mathrm{dz2}}\right]$ (2.32)

We check that this answer is correct in two ways. First we can re-calculate the Newman-Penrose coefficients and confirm that they are in the correct normal form.

 Spin > $\mathrm{NPCurvatureScalars}\left(\mathrm{W4},\mathrm{Dyad4}\right)$
 ${\mathrm{table}}\left(\left[{"Psi0"}{=}{0}{,}{"Psi1"}{=}{0}{,}{"Psi3"}{=}{0}{,}{"Psi4"}{=}{0}{,}{"Psi2"}{=}\frac{{25}}{{2}}\right]\right)$ (2.33)

This is the correct normal form since 

Second, we can calculate a type D Weyl spinor from this spinor dyad and check that the result coincides with the original spinor ${W}_{4}$.

 Spin > $\mathrm{W4Check}≔\mathrm{WeylSpinor}\left(\mathrm{Dyad4},"D",\mathrm{η4}\right)$
 ${\mathrm{W4Check}}{:=}{3}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}\frac{{9}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{-}\frac{{9}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}\frac{{1}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{-}\frac{{9}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}\frac{{1}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}\frac{{1}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{18}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{-}\frac{{9}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}\frac{{1}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}\frac{{1}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{18}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}\frac{{1}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{18}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{18}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{48}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.34)
 Spin > $\mathrm{W4}&minus\mathrm{W4Check}$
 ${0}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}$ (2.35)

Example 5. Type N

Define a rank 4 spinor ${W}_{5}.$

 Spin > $\mathrm{W5}≔\mathrm{DGzip}\left(\left[1,12,54,108,81\right],S,"plus"\right)$
 ${\mathrm{W5}}{:=}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{3}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{3}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{9}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{3}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{9}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{9}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{27}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{3}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{9}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{9}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{27}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{9}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{27}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{27}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{81}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.36)

Calculate the Newman-Penrose coefficients for ${W}_{5}$ with respect to the initial dyad basis .

 Spin > $\mathrm{NP5}≔\mathrm{NPCurvatureScalars}\left(\mathrm{W5},\left[\mathrm{dz1},\mathrm{dz2}\right]\right)$
 ${\mathrm{NP5}}{:=}{\mathrm{table}}\left(\left[{"Psi0"}{=}{81}{,}{"Psi1"}{=}{-}{27}{,}{"Psi3"}{=}{-}{3}{,}{"Psi4"}{=}{1}{,}{"Psi2"}{=}{9}\right]\right)$ (2.37)

Find the Petrov type of ${W}_{5}.$

 Spin > $\mathrm{PetrovType}\left(\mathrm{NP5}\right)$
 ${"N"}$ (2.38)

Compute an adapted contravariant spinor dyad for ${W}_{5}.$

 Spin > $\mathrm{SpinorDyadCon5}≔\mathrm{AdaptedSpinorDyad}\left(\mathrm{W5},"N"\right)$
 ${\mathrm{SpinorDyadCon5}}{:=}\left[{3}{}{\mathrm{D_z1}}{-}{\mathrm{D_z2}}{,}{\mathrm{D_z1}}\right]$ (2.39)

Here is the covariant form of the spinor dyad.

 Spin > $\mathrm{Dyad5}≔\mathrm{map}\left(\mathrm{RaiseLowerSpinorIndices},\mathrm{SpinorDyadCon5},\left[1\right]\right)$
 ${\mathrm{Dyad5}}{:=}\left[{\mathrm{dz1}}{+}{3}{}{\mathrm{dz2}}{,}{\mathrm{dz2}}\right]$ (2.40)

We check that this answer is correct in two ways. First we can re-calculate the Newman-Penrose coefficients and confirm that they are in the correct normal form.

 Spin > $\mathrm{NPCurvatureScalars}\left(\mathrm{W5},\mathrm{Dyad5}\right)$
 ${\mathrm{table}}\left(\left[{"Psi0"}{=}{0}{,}{"Psi1"}{=}{0}{,}{"Psi3"}{=}{0}{,}{"Psi4"}{=}{1}{,}{"Psi2"}{=}{0}\right]\right)$ (2.41)

This is the correct normal form since $1.$

Second, we can calculate a type N Weyl spinor from this spinor dyad and check that the result coincides with the original spinor ${W}_{5}$.

 Spin > $\mathrm{W5Check}≔\mathrm{WeylSpinor}\left(\mathrm{Dyad5},"N"\right)$
 ${\mathrm{W5Check}}{:=}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{3}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{3}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{9}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{3}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{9}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{9}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{27}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{3}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{9}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{9}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{27}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{9}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{27}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{27}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{81}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.42)
 Spin > $\mathrm{W5}&minus\mathrm{W5Check}$
 ${0}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}$ (2.43)