mentioned above. We leave it to the reader to check the condition that $\beta_i - \beta_j\notin\Phi$ for each choice of $S$.

\[example1\] If $S$ consists of the roots $$\beta_1 = \alpha_1, \quad \beta_2 = r_0 \alpha_1, \quad 
\beta_3 = r_{-1} r_0 \alpha_1,$$ then $\cF_S$ is a rank 3 subalgebra with Cartan matrix $$\bbm 2 & -2 & -2\\-2 & 2 & -2\\-2 & -2 & 2\ebm$$ and Weyl group $W_S = T(\infty,\infty,\infty)$ of index 6 in $W$.

There are other hyperbolic triangles with all angles zero in Figure 1, for example, the one with sides labelled $\alpha_1$, $\alpha_0$ and $r_{-1} r_0 r_1 r_0 \alpha_1 = 3\alpha_1 + 4\alpha_0 + 4\alpha_{-1}$. There are also 6 images of the fundamental triangle in this one, so these three reflections generate a subgroup of index 6 in $W$. The Cartan matrix generated from these three roots, taken as simple, is the same as the one above. Is there an automorphism of $\cF$ which interchanges these two isomorphic subalgebras? Are these two index 6 subgroups of $W$ conjugate?

\[example2\] If $S$ consists of the roots $$\beta_1 = r_1 r_0 \alpha_1 = 3\alpha_1 + 2\alpha_0,$$ $$\beta_2 = r_0 r_{-1} r_1 r_0 \alpha_1 = 3\alpha_1 + 6\alpha_0 