real, all reorderings of these expressions are equivalent.)

The above theorem generalizes to any Lorentzian KM algebra, $\cA$, that is, one which is obtained from an affine algebra by the procedure of “over-extension”, attaching an additional node by one line only to the affine node in an affine Dynkin diagram. For any root $\alpha$ such that $(\alpha,\alpha)\neq 0$, define $\alpha^\vee = 2\alpha/(\alpha,\alpha)$. Then the Cartan integers which are the entries of the Cartan matrix of the rank $r+2$ algebra $\cA$ are given by $(\alpha_i,\alpha_j^\vee)$.

\[rankr1thm\] Let $\{\alpha_i\ | -1\leq i\leq r\}$ be the simple roots of a Lorentzian KM algebra, $\cA$, so $\alpha_1,\dots,\alpha_r$ are the simple roots of a finite dimensional simple Lie algebra, $\alpha_0$ is the affine simple root which generates an affine root system when included in the list, and $\alpha_{-1}$ satisfies $(\alpha_{-1},\alpha_{-1}) = 2$,$(\alpha_{-1},\alpha_0) = -1$, $(\alpha_{-1},\alpha_i) = 0$ for $1\leq i\leq r$. Write the affine null root $\delta = \sum_{j=0}^r n_j \alpha_j$ where $n_0 =1$. Fix any integer $m \geq 0$ and define $$\beta_0 = \alpha_{-1}, \quad
\beta_1 = \beta_1^{(m)} = m\delta + \alpha_1, \quad
\beta_j = \alpha_j \;\;\; {\rm for} \; 2\leq j\leq r.$$ Then the KM Lie subalgebra $\cA(\beta_0,\cdots,\beta_r)$ constructed in Theorem \[subrootsys\] has 