Example 1.

Use the command SimpleLieAlgebraData to obtain the Lie algebra data for the simple Lie algebra This is the 15-dimensional Lie algebra of trace-free, skew-Hermitian matrices.

Initialize the Lie algebra

The command StandardRepresentation will produce the actual matrices defining .  (This command only applies to Lie algebras constructed by the  procedure.)

The Lie algebra elements corresponding to the complex diagonal matrices define a Cartan subalgebra.

We check this is indeed a Cartan subalgebra using the Query command

Here is the root space corresponding to the root <I, I, -I>.

We check that the X is an eigenvector for the elements of the Cartan subalgebra.

The column vector <I, I, I> is not a root

Example 2.

Here is the full root space decomposition for the Lie algebra from Example 1.

The second calling sequence for simply converts the given root vector to a list and extracts the corresponding root space from the root space decomposition table.

DifferentialGeometry, CartanSubalgebra, GetComponents, Query, RootSpaceDecomposition, SimpleLieAlgebraData, SimpleLieAlgebraProperties, StandardRepresentation