r_1 = \bpm 1 & 0\\0 & -1\epm.$$ The root system of $\cF$ is $$\Phi = \{N\in S_2(\bZ)\ |\ \det(N) \geq -1\}$$ which is just the elements $N$ of the root lattice for which the norm squared is less than or equal to 2, $-2\det(N) \leq 2$. The real roots (Weyl conjugates of the simple roots) are $$\Phi_{real} = \{N\in S_2(\bZ)\ |\ \det(N) = -1\}$$ which lie on a single sheeted hyperboloid. The light-cone is the set of points $N\in S_2(\bR)$ where $\det(N) = 0$. All real roots have multiplicity one, and in the case of $\cF$, the same is true of all roots on the light-cone. This comes from $\cF_0$ (with simple roots $\alpha_0$ and $\alpha_1$) whose underlying finite dimensional Lie algebra is the rank one Lie algebra $sl_2$ of type $A_1$. Any light-cone root, $N\in\Phi$, satisfies $\det(N) = 0$ and is $W$ equivalent to a unique one on level 0 of the form $\bpm a &0\\0 &0\epm = -a(\alpha_0+\alpha_1)$ with $0\neq a\in\bZ$. The roots $\Phi$ decompose into the disjoint union of positive and negative roots, and we can distinguish these level $0$ light-cone roots easily according to whether $a$ is negative or positive. The roots on level 1 