Example 1

We calculate the Cartan subalgebra for  the 8-dimensional Lie algebra of 3x3 trace-free matrices.  The structure equations are obtained using the SimpleLieAlgebraData command.

Initialized the Lie algebra.

Find a Cartan subalgebra.

We can check that this subalgebra is Abelian (and hence nilpotent) and self-normalizing.

These properties can also be checked with the Query command

For the split real forms of  the  simple Lie algebras,  a Cartans subalgebra can always be found consisting of diagonal matrices in the standard representation.

Example 2

Other Cartan subalgebras for can be found with the second calling sequence.

Example 3

The Cartan subalgebra of a nilpotent Lie algebra g is g itself.  Retrieve the stucture equations for a nilpotent Lie algebra from the DifferentialGeometry  library.

Check that the algebra is nilpotent.

Example 4

We find the Cartan subalgebra for a solvable Lie algebra. Retrieve the structure equations for a solvable Lie algebra from the DifferentialGeometry library.

Check that the algebra is solvable.

Example 5.

We find the Cartan subalgebra for a Lie algebra with a non-trivial Levi decomposition. Retrieve the structure equations for such a Lie algebra from the DifferentialGeometry library.

Check that the Levi decomposition is non-trivial.

Calculate the Cartan subalgebra.

DifferentialGeometry,   LieAlgebras,  CartanMatrix, Query, RootSpaceDecomposition