@Ka4; @Ka5; @KaM1]). It is apparent from the Cartan matrix and Dynkin diagram that $\cF$ contains an affine subalgebra $\cF_0$ of type $A_1^{(1)}$ with simple roots $\alpha_0$, $\alpha_1$. The approach in [@FF] is based on the decomposition $$\cF = \bigoplus_{n\in\bZ} \cF_n$$ with respect to $\cF_0$, that is, with respect to the “level” which is the value of the central element of $\cF_0$. The feature of $\cF$ which first attracted the attention of AJF was its Weyl group, $W$, which is a reflection group $$\la r_{-1},r_0,r_1\ |\ r_{-1}^2 = r_0^2 = r_1^2 = 1, (r_{-1} r_0)^3 = 1, 
(r_{-1} r_1)^2 = 1\ra$$ isomorphic to the hyperbolic triangle group, $T(2,3,\infty)$ and to $PGL_2(\bZ)$. The action of $W$ shows that there are infinitely many subalgebras of type $A_1^{(1)}$ inside $\cF$, corresponding to cusps of the modular group, or to lines of null roots on the light-cone. However, these lines of null roots are all related by the Weyl group, and therefore the corresponding affine subalgebras of $\cF$ are all $W$-equivalent.

To give more details, let $H$ be the Cartan subalgebra of $\cF$ and let $H^*$ be its dual space, which has a basis consisting of the simple roots $\{\alpha_{-1}, \alpha_0,\alpha_1\}$, and a symmetric 