Editing Lie group decomposition (section)

==List of decompositions==
* The [[Jordan–Chevalley decomposition]] of an element in algebraic group as a product of semisimple and unipotent elements
* The [[Bruhat decomposition]] <math>G=BWB</math> of a [[semisimple algebraic group]] into double [[coset]]s of a [[Borel subgroup]] can be regarded as a generalization of the principle of [[Gauss–Jordan elimination]], which generically writes a matrix as the product of an upper triangular matrix with a lower triangular matrix—but with exceptional cases. It is related to the Schubert cell decomposition of [[Grassmannian]]s: see [[Weyl group]] for more details.
*The [[Cartan decomposition]] writes a semisimple real Lie algebra as the sum of eigenspaces of a [[Cartan involution]].<ref>{{cite book |last=Kleiner |first=Israel |title=A History of Abstract Algebra |publisher=Birkhäuser |year=2007 |isbn=978-0817646844 |editor1-last=Kleiner |editor1-first=Israel |location=Boston, MA |doi=10.1007/978-0-8176-4685-1 |mr=2347309}}</ref>
* The [[Iwasawa decomposition]] <math>G=KAN</math> of a semisimple group <math>G</math> as the product of [[compact group|compact]], abelian, and [[nilpotent group|nilpotent]] subgroups generalises the way a square real matrix can be written as a product of an [[orthogonal matrix]] and an [[upper triangular matrix]] (a consequence of [[Gram–Schmidt orthogonalization]]).
*The [[Langlands decomposition]] <math>P=MAN</math> writes a parabolic subgroup <math>P</math> of a Lie group as the product of semisimple, abelian, and nilpotent subgroups.
* The [[Levi decomposition]] writes a finite dimensional Lie algebra as a [[semidirect product]] of a [[solvable Lie algebra|solvable]] ideal and a [[semisimple Lie algebra|semisimple]] subalgebra.
* The [[LU decomposition]] of a dense subset in the general linear group. It can be considered as a special case of the [[Bruhat decomposition]].
* The [[Birkhoff factorization|Birkhoff decomposition]], a special case of the [[Bruhat decomposition]] for affine groups.