${\mathrm{\χ}}_0\=2\left( {\mathrm{\Φ}}_{00}{\mathrm{\Φ}}_{02}-{\mathrm{\Φ}}_{01}^2\right)$

${\mathrm{\χ}}_1\={\mathrm{\Φ}}_{00}{\mathrm{\Φ}}_{12}\+{\mathrm{\Φ}}_{02}{\mathrm{\Φ}}_{10}-2 {\mathrm{\Φ}}_{01}{\mathrm{\Φ}}_{11}$

${\mathrm{\χ}}_2\=\frac{1}{3}\left({\mathrm{\Φ}}_{00}{\mathrm{\Φ}}_{22}-2 {\mathrm{\Φ}}_{01}{\mathrm{\Φ}}_{21}\+{\mathrm{\Φ}}_{02}{\mathrm{\Φ}}_{20}\+4 {\mathrm{\Φ}}_{10}{\mathrm{\Φ}}_{12}-4 {\mathrm{\Φ}}_{11}^2\right)$

${\mathrm{\χ}}_3\={\mathrm{\Φ}}_{10}{\mathrm{\Φ}}_{22}-2 {\mathrm{\Φ}}_{11}{\mathrm{\Φ}}_{21}\+{\mathrm{\Φ}}_{12}{\mathrm{\Φ}}_{20}$

${\mathrm{\χ}}_4\=2\left({\mathrm{\Φ}}_{20}{\mathrm{\Φ}}_{22}-{\mathrm{\Φ}}_{21}^2\right)$

$E_{00}\=4\left({\mathrm{\Φ}}_{00}{\mathrm{\Φ}}_{11}-{\mathrm{\Φ}}_{01}{\mathrm{\Φ}}_{10}\right)$

$E_{01}\=2\left({\mathrm{\Φ}}_{00}{\mathrm{\Φ}}_{12}-{\mathrm{\Φ}}_{02}{\mathrm{\Φ}}_{10}\right)$

$E_{02}\=4\left({\mathrm{\Φ}}_{01}{\mathrm{\Φ}}_{12}-{\mathrm{\Φ}}_{02}{\mathrm{\Φ}}_{11}\right)$

$E_{10}\=2\left({\mathrm{\Φ}}_{00}{\mathrm{\Φ}}_{21}-{\mathrm{\Φ}}_{20}{\mathrm{\Φ}}_{01}\right)$

$E_{11}\= {\mathrm{\Φ}}_{00}{\mathrm{\Φ}}_{22}-{\mathrm{\Φ}}_{02}{\mathrm{\Φ}}_{20}$

$E_{12}\=2\left({\mathrm{\Φ}}_{01}{\mathrm{\Φ}}_{22}-{\mathrm{\Φ}}_{02}{\mathrm{\Φ}}_{21}\right)$

$E_{20}\=4\left({\mathrm{\Φ}}_{01}{\mathrm{\Φ}}_{21}-{\mathrm{\Φ}}_{20}{\mathrm{\Φ}}_{11}\right)$

$E_{21}\=2\left({\mathrm{\Φ}}_{10}{\mathrm{\Φ}}_{22}-{\mathrm{\Φ}}_{20}{\mathrm{\Φ}}_{12}\right)$

$E_{22}\=4\left({\mathrm{\Φ}}_{11}{\mathrm{\Φ}}_{22}-{\mathrm{\Φ}}_{12}{\mathrm{\Φ}}_{21}\right)$

${\mathrm{\Δ}}_{\mathrm{\Φ}}\={\mathrm{\Φ}}_{11}^2-{\mathrm{\Φ}}_{01}{\mathrm{\Φ}}_{21}$

$r_1\=2\left({\mathrm{\χ}}_2-2 {\mathrm{\Φ}}_{10}{\mathrm{\Φ}}_{12}\+2 {\mathrm{\Φ}}_{11}^2\right)$

$$\begin{gathered} \mathrm{\χ}{\'}_0^{}\=2\left(E_{00}{E}_{02}-E_{01}^2\right) \\ \mathrm{\χ}{\'}_1^{}\=E_{00}{E}_{12}\+\mathrm{E\_\_02}\mathrm{E\_\_10}-2 \mathrm{E\_\_01}\mathrm{E\_\_11} \end{gathered}$$

$$\begin{gathered} \mathrm{\χ}{\'}_2\=\frac{1}{3}\left(E_{00}{E}_{22}-2E_{01}{E}_{21}\+E_{02}{E}_{20}\+4E_{10}{E}_{12}-4E_{11}^2\right) \\ \mathrm{\χ}{\'}_3^{}\=E_{00}{E}_{22}-2 \mathrm{E\_\_11}\mathrm{E\_\_21}\+ \mathrm{E\_\_12}\mathrm{E\_\_20} \\ \mathrm{\χ}{\'}_4^{}\=2\left(E_{20}{E}_{22}-E_{21}^2\right) \end{gathered}$$

$$\begin{gathered} I_p\=\frac{1}{3}\left({\mathrm{\χ}}_0{\mathrm{\χ}}_4-4 {\mathrm{\χ}}_1{\mathrm{\χ}}_3\+3 {\mathrm{\χ}}_2^2\right) \\ \mathrm{r\_\_3}\={\mathrm{r\_\_1}}^2-\mathrm{I\_\_p} \end{gathered}$$

$J_p\={\mathrm{\χ}}_0{\mathrm{\χ}}_2{\mathrm{\χ}}_4\+2 {\mathrm{\χ}}_1{\mathrm{\χ}}_2{\mathrm{\χ}}_3-{\mathrm{\χ}}_0{\mathrm{\χ}}_3^2-{\mathrm{\χ}}_1^2{\mathrm{\χ}}_4-{\mathrm{\χ}}_2^3$

$H\=r_1{I}_p-J_p$

$r_2\=2\left({\mathrm{\Φ}}_{00}{\mathrm{\Φ}}_{11}{\mathrm{\Φ}}_{22}\+{\mathrm{\Φ}}_{01}{\mathrm{\Φ}}_{12}{\mathrm{\Φ}}_{20}\+{\mathrm{\Φ}}_{02}{\mathrm{\Φ}}_{10}{\mathrm{\Φ}}_{21}-{\mathrm{\Φ}}_{00}{\mathrm{\Φ}}_{12}{\mathrm{\Φ}}_{21}-{\mathrm{\Φ}}_{01}{\mathrm{\Φ}}_{10}{\mathrm{\Φ}}_{22}-{\mathrm{\Φ}}_{02}{\mathrm{\Φ}}_{11}{\mathrm{\Φ}}_{20}\right)$$k_i\=J_p\left(E_{\mathrm{ii}}-\raisebox{1ex}{$2 r_2{I}_p{\mathrm{\Phi }}_{\mathrm{ii}}$}\!\left/ \!\raisebox{-1ex}{$H$}\right.\right)\,\mathrm{ii}\=0\, 1\, 2\;$ no summing

${\mathrm{D}}_p\=J_p^2-I_p^3$

A. The Plebanski-Petrov type of the Ricci tensor $R$ is O.

Step A1. If all the Ricci scalars ${\mathrm{\Φ}}_{00}\={\mathrm{\Φ}}_{01}\= ..\. \= {\mathrm{\Φ}}_{22}\=0$, then the Segre type $S$ = [(1,111)].

Step A2. Otherwise, if ${\mathrm{\Δ}}_{\mathrm{\Φ}}\=0$, then S = [(2,11)].

Step A3. Otherwise, if $E_{00}\>0$, then S = [1,(111)].

Step A4. Otherwise, if $E_{00}\le 0$, then S = [(1,11)1].

B. The Plebanski-Petrov type of the Ricci tensor $R$ is N. 

Step B1. If $r_1\=0$, then $S$ = [(3,1)], otherwise $S$ = [(2,1)1].

C. The Plebanski-Petrov type of the Ricci tensor $R$ is D.

Step C1. If $r_1\=0$, then $S=\left[Z\stackrel{\&conjugate0;}{Z}\left(11\right)\right]$.  

Step C2. If $r_1\ne 0$ and all the $E_{00}\=E_{01}\= ..\. \=E_{22}\=0$, then $S$ = [(1,1)(11)].

Step C3. If $r_1\ne 0$, $\mathrm{\χ\_\_0}\ne 0$ and $\mathrm{\χ}{\'}_0\=0$, then $S$ = [(2,11)].  

Step C4. If $r_1\ne 0$, ${\mathrm{\χ}}_0\=0$and $\mathrm{\χ}{\'}_2\=0$, then $S$ = [2,(11)].

Step C5. If  $H\&equals;0$, then $S$ = [2,(11)], while if $H<0$, then $S\&equals;\left[Z\stackrel{\&conjugate0;}{Z}\left(11\right)\right]$.

D. The Plebanski-Petrov type of the Ricci tensor $R$ is $\mathrm{III}$.

Step D1. The Segre type of $R$ is [3,1].

E. The Plebanski-Petrov type of the Ricci tensor $R$ tensor is II.

Step E1. Then Segre type of $R$ is [2, 11].

F. The Plebanski-Petrov type of the Ricci tensor $R$ tensor is I.  

Step F1. If ${\mathrm{D}}_p<0$, then $S$ =[1,111] while if ${\mathrm{D}}_p\&gt;0$, then $S$ = [$\bar{Z}\mathrm{Z11}$].