This algebra is the minimal rank hyperbolic KM algebra with both finite and affine KM subalgebras. Rank 2 hyperbolic KM algebras are also quite interesting ([@F; @KaM2]) because of their connections with real quadratic fields and Hilbert modular forms [@LM]. They have infinitely many $A_1$ subalgebras, infinitely many rank 2 hyperbolic KM subalgebras, and our new work shows that they contain Borcherds subalgebras as well. But our main inspiration has been the hyperbolic algebra $\cF$, which seems to be the simplest such algebra that incorporates all the essential features of higher rank examples. Further evidence that $\cF$ deserves most serious study is that it contains all the symmetric rank 2 hyperbolics, as well as many other rank 3 hyperbolics. From a physicist’s point of view, it is attractive because it may appear as a hidden symmetry in (an extension of) Einstein’s theory of gravity in four space-time dimensions (see [@DHN] for a review of recent developments and further references).

Many interesting results about the structure of $\cF$ were obtained in [@FF], as well as a relationship with Siegel modular forms of genus 2, but a complete understanding of the root multiplicities has remained elusive despite considerable further work ([@BKM1; @Ka3; 