bilinear form whose matrix with respect to this basis is the Cartan matrix above. That form is indefinite, and can be very conveniently described as follows. On the space $S_2(\bR)$ of $2 \x 2$ symmetric real matrices define a bilinear form by $$\bpm a & b\\b & c\epm \cdot 
\bpm a' & b'\\b' & c'\epm
= 2bb' - ac' - a'c$$ so the associated quadratic form is $$2b^2 - 2ac = -2 \det \bpm a & b\\b & c\epm.$$ Then the root lattice of $\cF$ is isometric to $S_2(\bZ)$ where the entries $a$, $b$ and $c$ are integers, and the weight lattice of $\cF$ is isometric to $S'_2(\bZ)$ where $a$ and $c$ are integers but $b\in \frac{1}{2}\bZ$. We make the correspondence explicit by choosing $$\alpha_{-1} = \bpm 1 & 0\\0 & -1\epm, \quad
\alpha_0 = \bpm -1 & -1\\-1 & 0\epm,\quad
\alpha_1 = \bpm 0 & 1\\1 & 0\epm.$$ The Weyl group action of $A \in PGL_2(\bZ)$ on $N = \bpm a & b\\b & c\epm$ is given by $A(N) = ANA^t$ and the explicit matrices which represent the three simple reflections are $$r_{-1} = \bpm 0 & 1\\1 & 0\epm, \quad
r_0 = \bpm -1 & 1\\0 & 1\epm,\quad
