structural:summarization
Abelian, nilpotent, and solvable

Analogously to abelian, nilpotent, and solvable groups, one can define abelian, nilpotent, and solvable Lie algebras.

A Lie algebra g {\displaystyle {\mathfrak {g}}} is abelian if the Lie bracket vanishes; that is, = 0 for all x and y in g {\displaystyle {\mathfrak {g}}}. In particular, the Lie algebra of an abelian Lie group (such as the group R n {\displaystyle \mathbb {R} ^{n}} under addition or the torus group T n {\displaystyle \mathbb {T} ^{n}}) is abelian. Every finite-dimensional abelian Lie algebra over a field F {\displaystyle F} is isomorphic to F n {\displaystyle F^{n}} for some n ≥ 0 {\displaystyle n\geq 0}, meaning an n -dimensional vector space with Lie bracket zero.

A more general class of Lie algebras is defined by the vanishing of all commutators of given length. First, the commutator subalgebra (or derived subalgebra) of a Lie algebra g {\displaystyle {\mathfrak {g}}} is {\displaystyle }, meaning the linear subspace spanned by all brackets {\displaystyle } with x, y ∈ g {\displaystyle x,y\in {\mathfrak {g}}}. The commutator subalgebra is an ideal in g {\displaystyle {\mathfrak {g}}}, in fact the smallest ideal such that the quotient Lie algebra is abelian. It is analogous to the commutator subgroup of a group.

A Lie algebra g {\displaystyle {\mathfrak {g}}} is nilpotent if the lower central series

becomes zero after finitely many steps. Equivalently, g {\displaystyle {\mathfrak {g}}} is nilpotent if there is a finite sequence of ideals in g {\displaystyle {\mathfrak {g}}},

such that a j / a j − 1 {\displaystyle {\mathfrak {a}}_{j}/{\mathfrak {a}}_{j-1}} is central in g / a j − 1 {\displaystyle {\mathfrak {g}}/{\mathfrak {a}}_{j-1}} for each j. By Engel's theorem, a Lie algebra over any field is nilpotent if and only if for every u in g {\displaystyle {\mathfrak {g}}} the adjoint endomorphism

is nilpotent.

More generally, a Lie algebra g {\displaystyle {\mathfrak {g}}} is said to be solvable if the derived series:

becomes zero after finitely many steps. Equivalently, g {\displaystyle {\mathfrak {g}}} is solvable if there is a finite sequence of Lie subalgebras,

such that m j − 1 {\displaystyle {\mathfrak {m}}_{j-1}} is an ideal in m j {\displaystyle {\mathfrak {m}}_{j}} with m j / m j − 1 {\displaystyle {\mathfrak {m}}_{j}/{\mathfrak {m}}_{j-1}} abelian for each j.

Every finite-dimensional Lie algebra over a field has a unique maximal solvable ideal, called its radical. Under the Lie correspondence, nilpotent (respectively, solvable) Lie groups correspond to nilpotent (respectively, solvable) Lie algebras over R {\displaystyle \mathbb {R} }.

For example, for a positive integer n and a field F of characteristic zero, the radical of g l (n, F) {\displaystyle {\mathfrak {gl}}(n,F)} is its center, the 1-dimensional subspace spanned by the identity matrix. An example of a solvable Lie algebra is the space b n {\displaystyle {\mathfrak {b}}_{n}} of upper-triangular matrices in g l (n) {\displaystyle {\mathfrak {gl}}(n)}; this is not nilpotent when n ≥ 2 {\displaystyle n\geq 2}. An example of a nilpotent Lie algebra is the space u n {\displaystyle {\mathfrak {u}}_{n}} of strictly upper-triangular matrices in g l (n) {\displaystyle {\mathfrak {gl}}(n)}; this is not abelian when n ≥ 3 {\displaystyle n\geq 3}.