The $I$ and $J$ Invariants

These are true invariants which do not depend on the choice of coordinates or the choice of tetrad, but PsiKadelia computes them via the $\Psi$'s:

$$I = \frac{1}{2}\left(2\,\Psi_0\,\Psi_4-8\,\Psi_1\,\Psi_3+6\,{\Psi_2}^2\right)$$

$$J = \det\left\vert \begin{array}{lcr} \Psi_0 & \Psi_1 & \Psi_2 \\ \Psi_1 & \Psi_2 & \Psi_3 \\ \Psi_2 & \Psi_3 & \Psi_4 \end{array}\right\vert.$$