.

(See CartanMatrix for the definition of  the vectors  )

Example 1.

We define 4 different Cartan matrices and calculate their standard forms and root type.  

Here are the standard forms, permutation matrices and root types.

For each example the second output is a permutation matrix which transforms the given input Cartan matrix to its standard form.

Example 2.

We define a 21-dimensional simple Lie algebra and calculate its root type.

Initialize this Lie algebra.

Find a Cartan subalgebra.

Find the root space decomposition.

Find the roots, positive roots and a choice of simple roots.

Find the Cartan matrix.

Transform the Cartan matrix to standard form. Here we use the second calling sequence. The command CartanMatrixToStandardForm now returns a permuted set of simple roots for which the Cartan matrix will be in standard form.

Check the result by re-calculating the Cartan matrix with respect to the permuted set of roots. We get the standard form immediately.

The root type of our 21-dimensional Lie algebra is

DifferentialGeometry, CartanMatrix, CartanSubalgebra,  PositiveRoots, RootSpaceDecomposition, SimpleRoots