This is probably easier to solve than my latest question and might be a useful lemma there.
Using the same notation as there, let $F(x,y,z)=0$ be a surface with degree $2$ in all $x,y,z$, and such that there exist $(x_1,y_1),(x_2,y_2)$, two different "special points" which fulfil $\forall_z F(x_1,y_1,z)=0$ etc., and $w=((x-x_3)(y-y_4))/((x-x_4)(y-y_3))$. Prove: Eliminating $x$ (and introducing $w$) gives some new surface $F'(w,y,z)=0$ with the same "signature" (also quadratic in all variables) exactly if $(x_3,y_3)=(x_1,y_1),(x_4,y_4)=(x_2,y_2)$ are indeed two different special points of $F$.
Bonus: Some relevance to the Euler Brick does have the fact that you can not necessarily repeat this with other variable pairs until the cows come home, since you might run out of pairs of special points, or, as bad, those points are not rational (which is not a complete no-go as $w$ might still produce a rational map), e.g. in the EB case soon $I$ and $\sqrt2$ pop up. So, what are the new special points of $F'$, if you know those of $F$? Does those transformations, pray tell, even form a group in the end?
