Example 1.

We test if certain subalgebras of are Cartan subalgebras. First define the standard matrix representation for as the space of trace-free matrices.

Calculate the structure equations for these matrices and initialize the resulting Lie algebra.

Let's check that is semi-simple.

Test to see if a list of vectors defines a Cartan subalgebra.

 Since  has 2 elements, this implies that the rank of   is 2. We can use this information to simplify checking that other subalgebras are Cartan subalgebras

Here is a 2-dimensional Abelian subalgebra which is not self-normalizing and therefore not a Cartan subalgebra.

Example 2.

The notion of a Cartan subalgebra is not restricted to semi-simple Lie algebras. We define a solvable Lie algebra and test to see if some subalgebras are Cartan subalgebras.

Any subalgebra which is an ideal cannot be a Cartan subalgebra.

DifferentialGeometry,  CartanSubalgebra, LieAlgebraData, Query[Ideal], Query[Solvable], Query[Subalgebra], Query[Semisimple]