How the square model of Fano 3-space, a.k.a. $PG(3, 2)$, embeds in the MOG?

Mathematics Asked by riccardoventrella on January 13, 2021

I’m studying deeply the Curtis’ MOG original article, i.e. a New combinatorial approach to $M_{24}$ and I’m still struggling a bit in reading the final 35 6×4 matrices representing the MOG.
Actually it’s clear to me they are split into a brick made by 2 1×4 columns and which are gotten collecting the 35 arrangements of a 8-set into couples of tetrads. Here the black and white squares are related to the $GF(2)$ entries.
The right part of the 6×4 matrix is 4×4 square, which contains 4 symbols per square. Actually this highly reminds me the $PG(3, 2)$ square model, as depicted here:
The square model of Fano 3-space

Since this is made by 35 entries too, I really think there’s a strong connection with the squares in the MOG, even if different. This is also suggested in many places by Cullinane in his Finite geometry site. The point is not clear to me how to interpret those 4 symbols. I guess they are taken from $GF(4)$ but I cannot see a clear mapping with the symbols.

I know from SPLAG Conway produced yet another representation of the MOG, using the Hexacode and $GF(4)$, but this approach was not present in the original seminal Curtis article, so I cannot see how those $GF(4)$ entries are gotten there.

Thank for the help.

EDIT: I’ve rephrased the title question to better representing my doubt.

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