and $r_{-1} r_0 \alpha_1 = \alpha_1 + 2\alpha_0 + 2\alpha_{-1}$. Since that triangle has all angles zero, and contains 6 copies of $\cD$, it means that reflections with respect to these three roots generate a subgroup of $W$ of index 6 isomorphic to the hyperbolic triangle group $T(\infty,\infty,\infty)$. Using Theorem \[subrootsys\], those roots determine a rank 3 hyperbolic subalgebra of $\cF$, inequivalent to any of the algebras in the series found at the end of the last section.

More generally, for any $n\geq 1$, let $S = \{\beta_1,\cdots,\beta_n\}$ be a subset of $n$ roots from $\Phi^+_{real}$, corresponding to reflection lines in the Poincaré disk, such that $\beta_i - \beta_j\notin\Phi$. Then the KM subalgebra $\cF_S = \cF(\beta_1,\cdots,\beta_n)$ determined by Theorem \[subrootsys\] has Weyl group, $W_S$, generated by the reflections $r_{\beta_1},\cdots,r_{\beta_n}$. The Cartan matrix of $\cF_S$ is $C_S = [(\beta_i,\beta_j)]$ since all real roots of $\cF$ are of squared length $2$. The fundamental domain $\cD_S$ of $W_S$ is a union of images of $\cD$, and the number of such images will be equal to the index of $W_S$ in $W$. Using Figure 1 we can find choices of $S$ which give either finite or infinite index sub-Weyl groups. Our first example was 