(1 - q^{4j-2})\right]
\frac{q^{-3}}{2}\left[ \prod_{j\geq 1} (1 + q^{2j-1}) - \prod_{j\geq 1} (1 - q^{2j-1})
- 2q \right]$$ was one of the main results of [@FF], but further work has shown the increasing complexity of higher levels as $\cF_0$-modules.

There are other ways in which one might naturally decompose $\cF$, for example, with respect to the finite dimensional subalgebra of type $A_2$ with simple roots $\alpha_{-1}$ and $\alpha_0$. In that decomposition the analogue of level of a root $\alpha = \sum a_i \alpha_i$ would be the coefficient $a_1$ of $\alpha_1$. Each graded piece of the decomposition would be a finite dimensional $A_2$ module, and it is not hard to use a computer to find the irreducible modules which occur and their multiplicities for quite a few levels[^2]. The question is whether there is a useful (recursive) description of those module multiplicities which sheds more light on the hyperbolic root multiplicities.

Indefinite subalgebras from subroot systems
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The main point of the present contribution is that indefinite KM algebras possess infinitely many non-isomorphic subalgebras, which are of indefinite type (in fact, KM subalgebras of equal rank with inequivalent Cartan matrices). Unlike finite or affine subalgebras, these subalgebras are themselves not yet 