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 combinations of Lie brackets of pairs of elements of . The derived series is the sequence of subalgebras If the derived series eventually arrives at the zero subalgebra, then the Lie algebra is called solvable. The derived series for Lie algebras is analogous to the derived series for commutator subgroups in group theory, and solvable Lie algebras are analogs of solvable groups. Any nilpotent Lie algebra is a fortiori solvable but the converse is not true. The solvable Lie algebras and the semisimple Lie algebras form two large and generally complementary classes, as is shown by the Levi decomposition. The solvable Lie algebras are precisely those that can be obtained from semidirect products, starting from 0 and adding one dimension at a time. A maximal solvable subalgebra is called a Borel subalgebra. The largest solvable ideal of a Lie algebra is called the radical. Characterizations Let be a finite-dimensional Lie algebra over a field of characteristic . The following are equivalent. (i) is solvable. (ii) , the adjoint representation of , is solvable. (iii) There is a finite sequence of ideals of : (iv) is nilpotent. (v) For -dimensional, there is a finite sequence of subalgebras of : with each an ideal in . A sequence of this type is called an elementary sequence. (vi) There is a finite sequence of subalgebras of , such that is an ideal in and is abelian. (vii) The Killing form of satisfies for all in and in . This is Cartan's criterion for solvability. Properties Lie's Theorem states that if is a finite-dimensional vector space over an algebraically closed field of characteristic zero, and is a solvable Lie algebra, and if is a representation of over , then there exists a simultaneous eigenvector of the endomorphisms for all elements . Every Lie subalgebra and quotient of a solvable Lie algebra are solvable. Given a Lie algebra and an ideal in it, is solvable if and only if both and are solvable. The analogous statement is true for nilpotent Lie algebras provided is contained in the center. Thus, an extension of a solvable algebra by a solvable algebra is solvable, while a central extension of a nilpotent algebra by