Let g be a simple Lie algebra, h a Cartan subalgebra, and  the root space decomposition of g with respect to h.  Let <$,$>  be the Killing form of g.  For each root , there are vectors and  such that and  These conditions uniquely determine The vector  can be computed using the command  RootToCartanSubalgebraElementH.

Let  be a set of simple roots for g.  Then the associated Cartan matrix is the  matrix with entries  < , / <,  >. The entries of the Cartan matrix are 0, 1, -1 or 2.  The Cartan matrix is independent of the choice of Cartan subalgebra h but is dependent upon the ordering of the simple roots in

The Cartan matrix is the fundamental invariant for semi-simple Lie algebras over C -- two complex semi-simple Lie algebras are isomophic if and only if their Cartan matrices are the same, modulo a permutation of the vectors in the Cartan subalgebra. The command CartanMatrixToStandardForm will transform a given Cartan matrix to a standard form.

The Cartan matrix encodes the re-construction of the root system of the Lie algebra from its simple roots. See PositiveRoots .

The information contained in the Cartan matrix is also encoded in the Dynkin diagram of the Lie algebra.  

The first calling sequence calculates the Cartan matrix of a Lie algebra from a set of simple roots and a root space decomposition.

The second calling sequence displays the standard form of the Cartan matrix for each possible root type of a simple Lie algebra.