\geq 1$. In the root system $\Phi$ of $\cF$ we have the positive root vectors $$\beta_0 = \alpha_{-1} = \bpm 1 & 0\\0 & -1\epm,$$ $$\beta_1 = \beta_1^{(m)} = m(\alpha_0 + \alpha_1) \pm \alpha_1 
= \bpm -m &\pm 1\\\pm 1 &0\epm.$$ Then the KM Lie subalgebra ${{\mathfrak{g}}}(\beta_0,\beta_1)$ constructed in Theorem \[subrootsys\] has Cartan matrix $$\bbm 2 & -m \\ -m & 2 \ebm$$ which is rank 2 hyperbolic for $m \geq 3$. To indicate the dependence on $m$ we will denote this subalgebra by $\cH(m)$.

The result follows from Theorem \[subrootsys\] because $\beta_0, \beta_1^{(m)} \in\Phi_{real}^+$ and $$\beta_1^{(m)} - \beta_0 =  m(\alpha_0 + \alpha_1) \pm \alpha_1 - \alpha_{-1}
= \bpm -m-1 &\pm 1\\\pm 1 &1\epm$$ has determinant $-m-2 < -2$ for $m\geq 1$, so is not a root of $\cF$.

While the real root vector $E_0$ of $\cH(m)$ may be taken to be the simple root generator $e_{-1}$ of $\cF$, the real root vector $E_1$ may be written as the $2m$-fold commutator $$E_1 = [e_1,[e_0,[e_1,\cdots  [e_1,[e_0 , e_1] \cdots ]]]$$ if $\beta_1^{(m)} = m(\alpha_0 + \alpha_1) + \alpha_1$, and as the commmutator $$E_1 = [e_0,[e_1,[e_0,\cdots  [e_0,[e_1 , e_0] \cdots ]]]$$ if $\beta_1^{(m)} = m(\alpha_0 + \alpha_1) - \alpha_1$. (Because $\beta_1^{(m)}$ is 