We give a method of determining the centralizer of an elliptic element in a real semisimple Lie algebra , in relation with the maximal compact subalgebra of and the compact dual of . Moreover, we determine a special central element (called the -element) of the isotropy subalgebra of each simple irreducible pseudo-Hermitian symmetric Lie algebra.
Nobutaka BOUMUKI. "Isotropy subalgebras of elliptic orbits in semisimple Lie algebras, and the canonical representatives of pseudo-Hermitian symmetric elliptic orbits." J. Math. Soc. Japan 59 (4) 1135 - 1177, October, 2007. https://doi.org/10.2969/jmsj/05941135