In my self-study of quadratic forms I am using J.W.S. Cassels' Rational quadratic forms (1978) as my main source of wisdom. For some time now I have been stuck in the chapter that covers ternary forms, i.e. forms in three variables, and I am kindly asking for help to get over this. You can find a link to the relevant chapter here, but I think it suffices to take a look at the following screenshot of page 304:
Cassels introduces an indefinite ternary form in (6.1), "indefinite" meaning that the thing takes on both positive and negative values, and then come the substitutions $X_j$ and $\xi_j$. The map $\pi$ then maps rational or real 3-tuples $x$ to $\xi = (\xi_1, \xi_2)$.
A little later, my big stumbling block arrives in the sentence that ends with (6.9): $T$ appears to be a real $3\times3$ matrix, and Cassels casually mentions that the mapping $x \mapsto Tx$ induces a fractional linear transformation of the plane. What exactly does that "induction" look like, i.e. what exactly is $T(\xi)$? A fractional linear transformation of the plane must be a mapping from $\mathbb{R}^2$ onto itself, and if $x \mapsto Tx$, which maps from $\mathbb{R}^3$ onto itself, is to play a role in this, we will very probably need a mapping from $\mathbb{R}^2$ to $\mathbb{R}^3$ to make the connection, but I cannot see any such mapping here. Can anybody help me with that?
I would greatly appreciate any help, especially as it certainly takes some time to get into the details of all this. If there is anybody around who feels like discussing this entire chapter about ternary forms with me I would be even more grateful.