Splitting Cartan Subalgebra
Over non-algebraically closed fields, not all Cartan subalgebras are conjugate. An important class are splitting Cartan subalgebras: if a Lie algebra admits a splitting Cartan subalgebra then it is called splittable, and the pair is called a split Lie algebra; over an algebraically closed field every semisimple Lie algebra is splittable. Any two splitting Cartan algebras are conjugate, and they fulfill a similar function to Cartan algebras in semisimple Lie algebras over algebraically closed fields, so split semisimple Lie algebras (indeed, split reductive Lie algebras) share many properties with semisimple Lie algebras over algebraically closed fields.
Over a non-algebraically closed field not every semisimple Lie algebra is splittable, however.
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