This means that if we choose a basis for a and extend to a basis  for h, then the components of a compact root  in the directions are 0.  If    determines the root space for  then  for  With respect to the standard Cartan algebras for the non-compact, simple matrix algebras we consider here, the compact roots are precisely those which are purely imaginary complex numbers.

Example 1.

We  find the compact roots for First we use the command SimpleLieAlgebraData to initialize the Lie algebra

For this example we use the command  SimpleLieAlgebraProperties to generate the various properties of that we need.

Here is the Cartan subalgebra.

Here is the Cartan subalgebra decomposition

We check that the restriction of the Killing form to the diagonal matrices   with imaginary entries is negative-definite. The restriction of the Killing form to the diagonal matrices  with real entries is positive-definite.

The second list of vectors in (2.3)  is therefore our subalgebra as described above.

Next we find  the positive roots.

The compact roots are:

Note that these roots all have purely imaginary components.

Example 2.

We find the compact roots for First we use the command SimpleLieAlgebraData to initialize the Lie algebra

We use the command SimpleLieAlgebraProperties to generate the various properties of that we need.

Here is the Cartan subalgebra.

Here is the Cartan subalgebra decomposition

The restriction of the Killing form to the diagonal matrices with imaginary entries is negative-definite. The restriction of the Killing form to the diagonal matrices  with real entries is positive-definite.

The second list of vectors in (2.3)  is therefore our subalgebra as described above.

Next we find the positive roots.

The compact roots are:

DifferentialGeometry,  CartanDecomposition, Cartan Involution, CartanSubalgebra, DynkinDiagram,  PositiveRoots, RootSpaceDecomposition, RestrictedRootSpaceDecomposition, SatakeDiagram,  SimpleLieAlgebraProperties, SimpleRoots