\det(C_{r-1}^{fin}).$$ Using the table of values of these determinants on page 53 of [@Kac], we find that $\det(C) < 0$ in all cases for $m\geq 2$, guaranteeing that the subalgebra will be indefinite.

For example, if the finite dimensional algebra with simple roots $\alpha_1,\dots ,\alpha_r$ is of type $A_r$, then it is easy to see that $\det(C) = 2(r+1) - m^2 r$, which will be negative when $m^2 > 2(r+1)/r$. For $r\geq 2$, this is true for $m \geq 2$, and for $r = 1$ this is true for $m\geq 3$. The $r = 1$ case is rather extreme since we only have $\beta_0 = \alpha_{-1}$ and $\beta_1 = m\delta + \alpha_1$ giving the rank 2 Cartan matrix already studied in Theorem \[rank2thm\].

The above subalgebras $\cA_{r+1}^{indef}(m)$ of $\cA$ for different values of $m\geq 2$ have inequivalent Cartan matrices. Therefore, $\cA$ possesses infinitely many nonisomorphic indefinite KM subalgebras corresponding to the infinitely many ways of slicing the forward light-cone in root space by time-like hyperplanes such that the hyperboloidal intersections become more and more steeply inclined with increasing $m$. For rank 3 all subalgebras were again hyperbolic, but the indefinite KM algebras obtained in this way for higher rank in 