A GCM is square matrix $A = (a_{ij})$ satisfying: (1) $a_{ij} \in \mathbf{Z}$ (2) $a_{ii} = 2$ for all $i$. (3) $a_{ij} \leq 0$ for $i \neq j$. (4) $a_{ij} = 0$ iff $a_{ji} = 0$. There is a standard notion of irreducibility among GCMs and the common term for it is "indecomposable". Now if you look at most books there is a standard basic classification of GCMs: An indecomposable  GCM is of 3 kinds: finite, affine and another category which can be described as all-other-GCMs-that-I-know-little-about-so-I-will-bundle-them-together. There is also another classification based on whether the GCM is "symmetrizable" or not (the finite and affine GCMs are <strike>symmetric and hence</strike> symmetrizable). 

What I am looking for is a smart classification of the GCMs that are not finite or affine. For example here is a number of questions: 
<list>
<ul> <b>[1]</b> Is the number of indefinite GCMs (not finite, not affine but symmetrizable) finite? As far as I know indefinite GCMs is a larger set than hyperbolic GCMs which are only finitely many.</ul> 
<ul> <b>[2]</b> Are all indefinite GCMs invertible? </ul> 
<ul> <b>[3]</b> If $A$ is indefinite and of size $N$ what is the biggest matrix of finite type that can appear as a principal minor in $A$? </ul>
</list>

__EDIT:__ corrected a mathematical mistake pointed out by Jim Humphreys, also improved the format as to make things more visible. Again suggested by Jim Humphreys.