with dual Cartan subalgebra Given a Lie algebra over a field of characteristic and a Lie algebra representation there is a decomposition related to the decomposition of the Lie algebra from its Cartan subalgebra. If we set with , called the weight space for weight , there is a decomposition of the representation in terms of these weight spaces In addition, whenever we call a weight of the -representation Classification of irreducible representations using weights But, it turns out these weights can be used to classify the irreducible representations of the Lie algebra . For a finite dimensional irreducible -representation there exists a unique weight with respect to a partial ordering on . Moreover, given a such that for every positive root there exists a unique irreducible representation This means the root system contains all information about the representation theory of 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. Cartan subgroup A Cartan subgroup of a Lie group is one of the subgroups whose Lie algebra is a Cartan subalgebra. The identity component of a subgroup has the same Lie algebra. There is no standard convention for which one of the subgroups with this property is called the Cartan subgroup, especially in the case of disconnected groups. A Cartan subgroup of a compact connected Lie group is a maximal connected Abelian subgroup (a maximal torus). Its Lie algebra is a Cartan subalgebra. For disconnected compact Lie groups there are several inequivalent definitions of a Cartan subgroup. The most common seems to be the one given by David Vogan, who defines a Cartan subgroup to be the group of elements that normalize a fixed maximal torus and fix the fundamental Weyl chamber. This is sometimes called the large Cartan subgroup. There is also a small Cartan subgroup,