the rank 3 hyperbolic KM algebra, $\cF$, which the most decisive example for understanding the subject. In section 3 we prove the key theorem which identifies a class of subalgebras of KM algebras by the choice of a set of positive real roots whose differences are not roots. We apply that theorem to the algebra $\cF$ to find inside it all simply laced rank 2 hyperbolic KM algebras, as well as an infinite series of inequivalent rank 3 indefinite KM algebras. We have generalizations of these results where the role of the algebra $\cF$ is played by an arbitrary rank $r+2$ Lorentzian KM algebra, $\cA$, and we find two infinite series of inequivalent indefinite KM subalgebras, one series of rank $r+1$ and one series of rank $r+2$. In section 4 we go back to the special case of $\cF$, and exploit the beautiful geometry of its Weyl group, $W$, which is the hyperbolic triangle group $T(2,3,\infty) = PGL_2(\bZ)$, through its action on the Poincaré disk model of the hyperbolic plane. This allows us to find, in principle, a large class of KM subalgebras of $\cF$ which are inequivalent to those already found in section 3. A complete classification of such 