In mathematics, a Lie algebra ${\displaystyle {\mathfrak {g}}}$ is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra of the Lie algebra ${\displaystyle {\mathfrak {g}}}$ is the subalgebra of ${\displaystyle {\mathfrak {g}}}$, denoted

that consists of all linear combinations of Lie brackets of pairs of elements of ${\displaystyle {\mathfrak {g}}}$. 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.

Let ${\displaystyle {\mathfrak {g}}}$ be a finite-dimensional Lie algebra over a field of characteristic 0. The following are equivalent.

Lie's Theorem states that if ${\displaystyle V}$ is a finite-dimensional vector space over an algebraically closed field of characteristic zero, and ${\displaystyle {\mathfrak {g}}}$ is a solvable Lie algebra, and if ${\displaystyle \pi }$ is a representation of ${\displaystyle {\mathfrak {g}}}$ over ${\displaystyle V}$, then there exists a simultaneous eigenvector ${\displaystyle v\in V}$ of the endomorphisms ${\displaystyle \pi (X)}$ for all elements ${\displaystyle X\in {\mathfrak {g}}}$.

A Lie algebra ${\displaystyle {\mathfrak {g}}}$ is called completely solvable or split solvable if it has an elementary sequence of ideals in ${\displaystyle {\mathfrak {g}}}$ from ${\displaystyle 0}$ to ${\displaystyle {\mathfrak {g}}}$. 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 ${\displaystyle 3}$-dimensional real Lie algebra of the group of Euclidean isometries of the plane is solvable but not completely solvable.