general are no longer hyperbolic even if the original subalgebra is hyperbolic.

The above construction can be further modified as follows to obtain subalgebras of rank $r+2$.

\[rankr2thm\] Let the notation be as in Theorem \[rankr1thm\]. Fix any integer $m \geq 1$ and define $$\gamma_{-1} = \alpha_{-1}, \quad
\gamma_0 = \gamma_0^{(m)} = (m-1)\delta + \alpha_0, \quad
\gamma_j = \alpha_j \;\;\; {\rm for} \; 1\leq j\leq r.$$ Then the KM Lie subalgebra $\cA(\gamma_{-1},\cdots,\gamma_r)$ constructed in Theorem \[subrootsys\] has Cartan matrix $$C = [(\beta_i,\beta_j^\vee)] = 
\bbm 2 & -m & 0 & \cdots & 0 \\ -m & 2 & * & \cdots & * \\
0 & * & 2 & \cdots & * \\ . &  &  &  &  \\ . &  &  &  &  \\ 0 & * & * & \cdots & 2 
\ebm$$ which is rank $r+2$ of indefinite type for $m\geq 1$, where the submatrix obtained by removing the first row and column of $C$ is the affine type Cartan matrix $C_{r+1}^{aff}$ determined by $\alpha_0,\dots,\alpha_r$. We also denote this subalgebra by $\cA_{r+2}^{indef}(m)$. The affine subalgebra with simple roots $\gamma_0,\cdots,\gamma_r$ has minimal null root $\sum_{j=0}^r n_j \gamma_j = m\delta$.

The proof is as in the previous theorem, 