(sometimes called the Cartan integers) where "ri" are the simple roots of the algebra. The entries are integral from one of the properties of roots. The first condition follows from the definition, the second from the fact that for formula_12 is a root which is a linear combination of the simple roots "ri" and "rj" with a positive coefficient for "rj" and so, the coefficient for "ri" has to be nonnegative. The third is true because orthogonality is a symmetric relation. And lastly, let formula_13 and formula_14. Because the simple roots span a Euclidean space, S is positive definite.