Why are there 12 complex spin coefficients in Newman-Penrose formalism and not 20?

The twelve coefficients are set up in the $B_{ab’}: $begin{pmatrix} A&DC&A end{pmatrix} and $A, D, C$ can have notation $E,F,G,H$ for $l,m,bar m,n$ where $phi = E,F,G,H$. Then all the twelve coefficient equations are set up thus: $A=EHphi-FGphi$, $D=EFphi$, and $C=HGphi$. So my question is why isn’t there an $I=EGphi$ and $J=HFphi$ in a matrix: $$begin{pmatrix}A&D&IC&A&DJ&C&A end{pmatrix}$$ I’m looking for more of a conceptual answer which could lead me away from error. Thanks.