$(ad\ F_i)^{1-C_{ji}}\ F_j = 0$ for all $i\neq j$, follow immediately since $-(1 - C_{ji})\beta_i - \beta_j$ is not in $\Phi$. This shows that all of the Serre relations determined by the Cartan matrix $C$ are satisfied by the generators given above.

Our first application of this theorem is to show that the rank 3 algebra $\cF$ contains all the simply laced rank 2 hyperbolic KM algebras as subalgebras. The decomposition of $\cF$ with respect to its affine algebra $\cF_0$ corresponded to slicing with respect to planes parallel to an edge of the light-cone, and the affine Weyl group acted on those planes by moving roots along parabolas. Decomposition with respect to the finite dimensional algebra $A_2$ corresponds to slices which intersect the light-cone in circles or ellipses. So it should come as no surprise that there should be subalgebras whose decompositions correspond to slices which intersect the light-cone in hyperbolas. Consequently we will “locate” the rank 2 hyperbolic KM algebras by identifying their simple root systems inside the root system of $\cF$. There are two equally good choices for each algebra, distinguished in the theorem below by a choice of plus or minus sign.

\[rank2thm\] Fix any integer $m 