Journal of Lie Theory 27 (2017), No. 4, 1027--1032
Copyright Heldermann Verlag 2017
Commutators and Cartan Subalgebras in Lie Algebras of Compact Semisimple Lie Groups
Dept. of Mathematics and Statistics, Notre Dame University, Louaize, Zouk Mikael, Lebanon
Dept. of Mathematics, Faculty of Arts and Sciences, American University of Beirut, Beirut, Lebanon
Short proofs are given of the following facts concerning the Lie algebra g of a compact semisimple Lie group.
(1) Any element in g is a commutator bracket of some two elements of g.
(2) Given a Cartan subalgebra h of g, there exists a Cartan subalgebra h' which is orthogonal to h.
Moreover, as a Corollary, we obtain the known fact that any element in g is conjugate to some element in the orthogonal complement of h.
Keywords: Semisimple Lie Algebras, commutators, Goto's Theorem, Cartan subalgebras.
MSC: 22E60, 20F12
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