I'm trying to figure out a proof in Lam's book on quadratic forms (he uses this to define a hyperbolic plane). He states that if $(V,q)$ is a 2-dimensional quadratic space over $F$, then the following are equivalent:
- $V$ is regular and isotropic.
- $V$ is regular and $\textrm{disc}(V)=-1\cdot F^{\times 2}$.
- $V$ is isometric to $\langle 1,-1\rangle$.
- $V$ corresponds to the equivalence class of the binary quadratic form $X_1X_2$.
I have trouble understanding his proof that $(2)\Rightarrow (3)$. He starts by stating that "We clearly have the diagonilization $V\simeq \langle a,-a\rangle$". I just don't see why this is clear assuming (2).