subalgebras is beyond the scope of this paper, but we give a number of representative examples to illustrate the idea. We also discuss how the geometry of the Poincaré disk gives information about the generators and relations defining the Weyl groups of these subalgebras, and how they are related to the Weyl group $W$. We then use this geometrical picture to relate the Weyl groups of the series of rank 3 indefinite subalgebras of $\cF$ found in section 3 to $W$. (Thanks to Tadeusz Januszkiewicz for suggesting and explaining this geometrical way of finding subgroups of $W$.) In section 5 we give a method of finding Borcherds generalized KM algebras inside $\cF$, inside and Lorentzian algebra $\cA$, and even inside rank 2 hyperbolics.

The rank 3 algebra $\cF$
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The canonical example of a hyperbolic KM algebra is the rank 3 algebra $\cF$ studied in [@FF], whose Cartan matrix is $$\bmatrix 2 & -1 & 0 \\ -1 & 2 & -2 \\ 0 & -2 & 2 \endbmatrix\ ,$$ whose simple roots are $\alpha_{-1}$, $\alpha_0$, $\alpha_1$, and whose Dynkin diagram is $$\bullet\underset{\hskip -10pt\alpha_{-1}} \ \hbox{-----} \ 
\bullet\underset{\hskip -10pt\alpha_{0}}\   
\vbox{\hbox{-----}\vskip -10pt \hbox{-----}
\vskip -10pt\hbox{-----} \vskip -10pt\hbox{-----}\vskip -3pt}\   
\bullet\underset{\hskip -10pt\alpha_{1}}\ .$$ 