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 ~~symmetric and hence~~ 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:

**[1]**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.

**[2]**Are all indefinite GCMs invertible?

**[3]**If $A$ is indefinite and of size $N$ what is the biggest matrix of finite type that can appear as a principal minor in $A$?

**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.

Lie Algebras of Finite and Affine Typeby R.W. Carter, the off-diagonal entries of a GCM are allowed at first to be arbitrary complex numbers. See Carter's Chapters 14 and 15 for his viewpoint on classification of GCMs and related Lie algebras. Other sources, such as the 1995 Wiley bookLie Algebras with Triangular Decompositionsby Moody-Pianzola, have different emphases. But GCMs mainly come up in Lie algebra theory. – Jim Humphreys Feb 15 '11 at 14:27