Example 1.

We consider the Lie algebra This is the 24-dimensional real Lie algebra of  6×6 complex matrices  which are trace-free and skew-Hermitian with respect to the quadratic form  . We use the command SimpleLieAlgebraData to initialize this Lie algebra.

We use the command SimpleLieAlgebraProperties to obtain the Cartan subalgebra, the root space decomposition, and the simple roots.

The result is a table. Here is the Cartan subalgebra for

Here is the root space decomposition for

Here are the positive roots.

Let us find  where  is the first root  (2.4) 

We check that  is in the Cartan subalgebra.

Here are the root spaces for and

We check that defines a Lie subalgebra.

If we scale the vectors X and Y then the structure equations take the standard form for . 

Example 2.

We illustrute how to use RootToCartanSubalgebraElementH to calculate the Cartan matrix for We first calculate the  for the simple roots .

Then we calculate the Killing form , restricted to subspace [

The Cartan matrix is given by normalizing the entries of

The Lie algebra is a rank 5 simple Lie algebra of type "A". The matrix in (2.15)  is therefore correct.

DifferentialGeometry,  CartanMatrix, Killing,  LieAlgebraData, RootSpace, SimpleLieAlgebraData, SimpleLieAlgebraProperties