${\mathrm{\Δ}}_{c }\= \{ \mathrm{\α} \in \mathrm{\Δ} \| {\mathrm{\α}}_{\|\mathrm{\𝔞}} \= 0\}\.$

This means that if we choose a basis $\left\{a_1\, a_{2 }\, .. \. \,a_s\right\}$for a and extend to a basis $\left\{a_1\, a_{2 }\, .. a_s\, h_{s\+1}\, ..\. \, h_m\right\}$ for h, then the components of a compact root ${\mathrm{\α}}_{}$ in the $\left\{a_1\, a_{2 }\, ..\. \,a_s\right\}$directions are 0. If $x \in \mathrm{\𝔤}$ determines the root space for ${\mathrm{\α}}_{}\,$then $\mathrm{ad}\left(a_i\right)\left(x\right) \= 0$ for $i \= 1..\. s\.$ 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 $\mathrm{su}\left(5\, 2\right)\.$First we use the command SimpleLieAlgebraData to initialize the Lie algebra$\mathrm{su}\left(5\, 2\right)\.$

For this example we use the command SimpleLieAlgebraProperties to generate the various properties of $\mathrm{su}\left(5\, 2\right)$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 $T$ with imaginary entries is negative-definite. The restriction of the Killing form to the diagonal matrices $A$ with real entries is positive-definite.

The second list of vectors in (2.3)  is therefore our subalgebra $\mathrm{\𝔞}\mathit{\,}\mathit{ }$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 $\mathrm{sp}\left(4\, 4\right)\.$First we use the command SimpleLieAlgebraData to initialize the Lie algebra$\mathrm{sp}\left(4\, 4\right)\.$

We use the command SimpleLieAlgebraProperties to generate the various properties of $\mathrm{sp}\left(4\,4\right)$that we need.

Here is the Cartan subalgebra.

Here is the Cartan subalgebra decomposition

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

The second list of vectors in (2.3)  is therefore our subalgebra $\mathrm{\𝔞}\mathit{\,}\mathit{ }$as described above.

Next we find the positive roots.

The compact roots are: