mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, hence the name. More concretely, a Lie algebra is reductive if it...

mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero...

of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms...

called reductive if its adjoint representation is semisimple. Every reductive Lie algebra is isomorphic to the product of an abelian Lie algebra and a...

Simple algebraic groups and (more generally) semisimple algebraic groups are reductive. Claude Chevalley showed that the classification of reductive groups...

used to read off the list of simple Lie algebras and Riemannian symmetric spaces. Together with the commutative Lie group of the real numbers, R {\displaystyle...

matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra. On the other hand, Lie groups with isomorphic Lie algebras...

maximal toral Lie subalgebra of a finite-dimensional semisimple Lie algebra, or more generally of a finite-dimensional reductive Lie algebra,[citation needed]...

triangularizable. If a Lie algebra admits a splitting, it is called a splittable Lie algebra. Note that for reductive Lie algebras, the Cartan subalgebra...

Cartan algebras in semisimple Lie algebras over algebraically closed fields, so split semisimple Lie algebras (indeed, split reductive Lie algebras) share...