Commutators and Cartan Subalgebras in Lie Algebras of Compact Semisimple Lie Groups

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 h ⊥ .