# Category: Properties of Lie algebras

Abelian Lie algebra
No description available.
Split Lie algebra
In the mathematical field of Lie theory, a split Lie algebra is a pair where is a Lie algebra and is a splitting Cartan subalgebra, where "splitting" means that for all , is triangularizable. If a Lie
Nilpotent Lie algebra
In mathematics, a Lie algebra is nilpotent if its lower central series terminates in the zero subalgebra. The lower central series is the sequence of subalgebras We write , and for all . If the lower
Compact Lie algebra
In the mathematical field of Lie theory, there are of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition includes
Reductive Lie algebra
In mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, whence the name. More concretely, a Lie algebra is reductive if it is a direct sum of a semisimple Lie
Toral subalgebra
In mathematics, a toral subalgebra is a Lie subalgebra of a general linear Lie algebra all of whose elements are semisimple (or diagonalizable over an algebraically closed field). Equivalently, a Lie
Free Lie algebra
In mathematics, a free Lie algebra over a field K is a Lie algebra generated by a set X, without any imposed relations other than the defining relations of alternating K-bilinearity and the Jacobi ide
Semisimple Lie algebra
In 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 proper ideals). Throughout the article
Solvable Lie algebra
In mathematics, a Lie algebra is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra of the Lie algebra is the subalgebra of , denoted that consists of all linear