$G_0\=2\left({\mathrm{\Ψ}}_0{\mathrm{\Ψ}}_2-{\mathrm{\Ψ}}_1^2\right)$

$G_1\=2\left({\mathrm{\Ψ}}_0{\mathrm{\Ψ}}_3-{\mathrm{\Ψ}}_1{\mathrm{\Ψ}}_2\right)$

$G_2\={\mathrm{\Ψ}}_0^2\+{\mathrm{\Ψ}}_0{\mathrm{\Ψ}}_4-2 {\mathrm{\Ψ}}_1{\mathrm{\Ψ}}_3$

$G_3\={\mathrm{\Ψ}}_1{\mathrm{\Ψ}}_4-{\mathrm{\Ψ}}_2{\mathrm{\Ψ}}_3$

$G_4\=2\left({\mathrm{\Ψ}}_2{\mathrm{\Ψ}}_4-{\mathrm{\Ψ}}_3^2\right)$

$G_5\=2\left({\mathrm{\Ψ}}_1{\mathrm{\Ψ}}_3-{\mathrm{\Ψ}}_2^2\right)$

$I\=G_2-G_5$

$J\=-{\mathrm{\Ψ}}_3{G}_1\+\frac{1}{2}\left({\mathrm{\Ψ}}_2{G}_5\+{\mathrm{\Ψ}}_4{G}_0\right)$

 If $I^3\=27 J^2$ then $\mathrm{\λ}$ is determined by ${\mathrm{\λ}}^2\=\frac{1}{3}I$ and ${\mathrm{\λ}}^3\=-J$and

$M_r\=G_r\+{\mathrm{\λ\Ψ}}_r \, r\=0\,1\,2\,3\,4$

 $L\=G_2\+2 G_5\+3 {\mathrm{\λ\Ψ}}_2$

Step 1. If ${\mathrm{\Ψ}}_0\={\mathrm{\Ψ}}_1\={\mathrm{\Ψ}}_2\={\mathrm{\Ψ}}_3\={\mathrm{\Ψ}}_4$=0, then the Petrov type is O.

Step 2. Otherwise, if $G_0\=G_1\=G_2\=G_3\=G_4\=0$, then the Petrov type is N.

Step 3. Otherwise, if $I\=J\=0$, then the Petrov type is III.

Step 4. Otherwise, if $I^3\ne 27 J^2$, then the Petrov type is I.

Step 5. If $I^3\=27 J^2$ and $M_r\=0$ for $r\=0\,1\,2\,3\,4$ and $G_2\+2 G_5\+3 {\mathrm{\λ\Ψ}}_2\=0$, then the Petrov type is D.

Step 6. Otherwise, the Petrov type is II.