+ 2\alpha_{-1},$$ $$\beta_3 = r_{-1} r_0 r_1 r_0 \alpha_1 = 3\alpha_1 + 4\alpha_0 + 4\alpha_{-1},$$ then $\cF_S$ is a rank 3 subalgebra with Cartan matrix $$\bbm 2 & -10 & -10\\-10 & 2 & -10\\-10 & -10 & 2\ebm$$ and Weyl group $W_S = T(\infty,\infty,\infty)$ of infinite index in $W$ because there are no relations between the generating reflections and the area enclosed by the reflection lines contains an infinite number of copies of $\cD$.

\[example3\] If $S$ consists of the roots $$\beta_1 = \alpha_1,\qquad  
\beta_2 = r_0 r_1 r_{-1} r_0 \alpha_1 = 3\alpha_1 + 6\alpha_0 + 2\alpha_{-1},$$ $$\beta_3 = r_0 r_{-1} r_0 r_1 r_0 \alpha_1 = 3\alpha_1 + 6\alpha_0 + 4\alpha_{-1},$$ then $\cF_S$ is a rank 3 subalgebra with Cartan matrix $$\bbm 2 & -6 & -6\\-6 & 2 & -2\\-6 & -2 & 2\ebm$$ and Weyl group $W_S = T(\infty,\infty,\infty)$ of infinite index in $W$ because there are no relations between the generating reflections and the area enclosed by the reflection lines contains an infinite number of copies of $\cD$.

\[example4\] If $S$ consists of the roots $$\beta_1 = r_0 \alpha_{-1} = \alpha_0 + \alpha_{-1},\qquad 
\beta_2 = r_1 \alpha_0 = 2\alpha_1 + \alpha_0,$$ $$\beta_3 = r_0 r_1 r_0 