Ideals, solvability, and Levi decomposition

A subspace I is an ideal when [g,I] is contained in I. This is stronger than being a subalgebra, which only requires [I,I] to stay in I. The ideal condition makes a quotient bracket on g/I well defined. It is also exactly invariance of I under the adjoint representation. An invariant subspace in some other representation is a different object, so a preserved coordinate line in the defining matrix representation does not by itself identify an ideal of the matrix Lie algebra.

The derived series begins with g and repeatedly brackets the current space with itself. Solvability means that this series eventually becomes zero. The lower-central series instead brackets the current space with all of g; its eventual vanishing defines nilpotence. Every nilpotent Lie algebra is solvable, but the converse fails. The distinction concerns how brackets are nested, not whether selected matrices in a realization happen to be nilpotent. A Lie algebra can contain semisimple and nilpotent operators while its derived series still terminates.

The radical r is the largest solvable ideal. In finite dimension, sums of solvable ideals remain solvable, so such a largest ideal exists. The quotient g/r is semisimple, meaning it has no nonzero solvable ideal. Levi's theorem in characteristic zero states that a semisimple subalgebra s maps isomorphically onto this quotient and complements r as a vector space. The theorem is an existence result about a subalgebra, not a claim that any arbitrary linear complement closes under brackets.

The bracket between s and r can be nonzero, producing a semidirect product. The complement s need not be an ideal and is generally not canonical, whereas r is intrinsic. Characteristic-zero conjugacy results relate Levi complements by suitable inner automorphisms, but do not select a preferred one without additional structure. This lesson uses the existence and semidirect interpretation, rather than attempting the cohomological proof. In positive characteristic or infinite dimension the same statement cannot simply be quoted unchanged.

A finite-dimensional complex Lie algebra is reductive when its radical equals its center. In that case the radical commutes with a Levi complement and the decomposition becomes a direct product of a central abelian algebra and a semisimple algebra. The algebra gl(n,C) is the basic example. Semisimple means the radical is zero, so its center is also zero; reductive permits a central part. These definitions should not be inferred merely from diagonalizability of one selected matrix or complete reducibility in one selected representation.

For a genuine semidirect example, let sl(2,C) act on its standard two-dimensional module and let that module be an abelian ideal. The resulting algebra has a two-dimensional radical and a nontrivial cross-action. Its semisimple quotient is sl(2,C), yet the whole algebra is not reductive because the radical is not central. This example explains why a quotient classification does not reconstruct the original bracket: one must also know the action on the radical and, before splitting, the extension data.