A Cartan matrix is a square integer matrix who elements (A_(i j)) satisfy the following conditions. 1.A_(i j) is an integer, one of {-3, -2, -1, 0, 2}. 2.A_(i i) = 2 the diagonal entries are all 2. 3.A_(i j)<=0 off of the diagonal. 4.A_(i j) = 0 iff A_(j i) = 0. 5. There exists a diagonal matrix D such that D A D^(-1) gives a symmetric and positive definite quadratic form.
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