but the submatrix obtained by removing the first row and column of $C$ is the affine type Cartan matrix $C_{r+1}^{aff}$ determined by $\alpha_0,\dots,\alpha_r$, and $C_r^{fin}$ is the finite type Cartan matrix obtained from $C_{r+1}^{aff}$ by deleting its first row and column, so $$\det(C) = 2\ \det(C_{r+1}^{aff}) - m^2\ \det(C_r^{fin}).$$ Since $\det(C_{r+1}^{aff}) = 0$, and $\det(C_r^{fin}) > 0$, we find that $\det(C) < 0$ in all cases for $m\geq 1$, guaranteeing that the subalgebra will be indefinite. Of course, the case when $m = 1$ just gives the original algebra $\cA$, so the only new content is for $m\geq 2$.

For the algebra $\cF$, and for $m \geq 2$, this procedure yields infinitely many inequivalent rank 3 subalgebras with the Cartan matrices $$\bbm 2 & -m & 0\\ -m & 2 & -2\\ 0 & -2 & 2 \ebm$$ not previously known to exist inside of $\cF$. In the next section we will explore the geometrical meaning of these subalgebras using the Weyl group of $\cF$.

Indefinite subalgebras of $\cF$ from the Weyl group
===================================================

A beautiful geometrical way to find subalgebras of $\cF$ is to examine the action of the Weyl group, $W$, which preserves each surface of fixed determinant. 