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 SimpleLieAlgebraData 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 RootSpace simply converts the given root vector to a list and extracts the corresponding root space from the root space decomposition table.
