The algebra is systematically incorrect throughout the article. In airplane space:

    Aga = Gag   (wrong!)

should be:

   Aga = reciprocal of Gag

since the A and G planes always have inverse perspective of each other. (Imagine both of them starting with a world at A=G=I, where I is facile to face with the opposing plane; but any pair of inverse perspectives works.):

  I = identity 

  x' x = 1 (x' is reciprocal  (inverse)of x)

  A' = G  <=>   AG = A A' = I

  a' = g

  A_2 = Aga = (Gag)' = (G_2)' 

     <=>  (A_2)(G_2) = (Aga)(gaG) = A g a  a' g' A' =  I

I'm page number space, the article's algebra is correct (but that's algebra, not a matrix transformation), because the the opposing books are reciprocals of each other in airplane space. This also explains why an A book can't fight an A book.

Since A=G and Aga=Gag in page number space, this also shows that ga=ag in page number space. ga (and ag, and a, an g) can be seen as transformations (turn n pages, where n can be positive or negative). And to make this a proper algebra (group action), we can say that A and G are also transformations, applied to an arbitrary starting point in the books.

In the end, the page turns are transformation matrixes (because all algebraic groups have a matrix represention), but it's not the 3D airplane transformation matrix.

Homework: create a matrix-multiplication representation for the algebra of adding integers.