a nilpotent algebra is nilpotent. A solvable nonzero Lie algebra has a nonzero abelian ideal, the last nonzero term in the derived series. If are solvable ideals, then so is . Consequently, if is finite-dimensional, then there is a unique solvable ideal containing all solvable ideals in . This ideal is the radical of . A solvable Lie algebra has a unique largest nilpotent ideal , called the nilradical, the set of all such that is nilpotent. If is any derivation of , then . Completely solvable Lie algebras A Lie algebra is called completely solvable or split solvable if it has an elementary sequence{(V) As above definition} of ideals in from to . A finite-dimensional nilpotent Lie algebra is completely solvable, and a completely solvable Lie algebra is solvable. Over an algebraically closed field a solvable Lie algebra is completely solvable, but the -dimensional real Lie algebra of the group of Euclidean isometries of the plane is solvable but not completely solvable. A solvable Lie algebra is split solvable if and only if the eigenvalues of are in for all in . Examples Abelian Lie algebras Every abelian Lie algebra is solvable by definition, since its commutator . This includes the Lie algebra of diagonal matrices in , which are of the formfor . The Lie algebra structure on a vector space given by the trivial bracket for any two matrices gives another example. Nilpotent Lie algebras Another class of examples comes from nilpotent Lie algebras since the adjoint representation is solvable. Some examples include the upper-diagonal matrices, such as the class of matrices of the formcalled the Lie algebra of strictly upper triangular matrices. In addition, the Lie algebra of upper diagonal matrices in form a solvable Lie algebra. This includes matrices of the formand is denoted . Solvable but not split-solvable Let be the set of matrices on the formThen is solvable, but not split solvable. It is isomorphic with the Lie algebra of the group of translations and rotations in the plane. Non-example A semisimple Lie algebra is never solvable since its radical , which is the largest solvable ideal in , is trivial. page 11 Solvable Lie groups Because the term "solvable" is also used for solvable groups in group theory, there are several possible definitions of solvable Lie group. For a Lie group , there is termination of the usual derived series of the