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 $r+1$ of indefinite type for $m\geq 2$, where the submatrix obtained by removing the first row and column of $C$ is the finite type Cartan matrix $C_r^{fin}$ determined by $\alpha_1,\dots,\alpha_r$. We also denote this subalgebra by $\cA_{r+1}^{indef}(m)$.

As in Theorem \[rank2thm\], for $m \geq 0$, the roots $\beta_i$, $0\leq i\leq r$, satisfy the conditions of Theorem \[subrootsys\] and therefore determine a KM subalgebra of $\cA$ whose Cartan matrix is as shown because $(\delta,\delta) = 0$, $(\delta,\alpha_i) = 0$ for $0\leq i\leq r$, and $(\delta,\alpha_{-1}) = -1$. Note that the submatrix obtained by removing the first row and column of $C$ is the finite type Cartan matrix $C_r^{fin}$ determined by $\alpha_1,\dots,\alpha_r$. Let $C_{r-1}^{fin}$ be the finite type Cartan matrix obtained from $C_r^{fin}$ by deleting its first row and column. Then we see that $$\det(C) = 2\ \det(C_r^{fin}) - m^2\ 