Levi decomposition

Remarks

A subalgebra $\mathfrak s$ as in the statement is a Levi subalgebra of $\mathfrak g$. By a theorem of Malcev, any two Levi subalgebras are conjugate by an automorphism $\exp(\operatorname{ad} z)$ with $z$ in the nilradical of $\mathfrak g$.

Idea of the proof: by induction on $\dim \mathfrak r$ one reduces to the case where $\mathfrak r$ is abelian and is a minimal nonzero ideal. Then $\mathfrak g$ is an extension of the semisimple Lie algebra $\mathfrak g / \mathfrak r$ by the module $\mathfrak r$. Such an extension splits because $H^2(\mathfrak g / \mathfrak r, \mathfrak r) = 0$ by the second of Whitehead's lemmas, and a splitting is a Levi subalgebra.

Example: let $\mathfrak g = \mathfrak{sl}_2(k) \oplus k^2$ with the bracket $[(x, v), (y, w)] = ([x, y], xw - yv)$, where $\mathfrak{sl}_2(k)$ is the special linear Lie algebra. Here $k^2$ is an abelian ideal and $\mathfrak g / k^2 \cong \mathfrak{sl}_2(k)$ is semisimple, so $\mathfrak r = k^2$ and $\mathfrak{sl}_2(k)$ is a Levi subalgebra.

Going deeper

For a student

The Levi decomposition reduces the structure theory of finite-dimensional Lie algebras in characteristic $0$ to two parts, solvable algebras and semisimple algebras. The two parts are glued by an action of the semisimple part on the radical. Over $\C$, semisimple algebras are classified by Dynkin diagrams, while solvable ones have no such classification. The analogue for simply connected Lie groups is the Levi–Malcev decomposition of a group into a solvable normal subgroup and a semisimple subgroup. The proof is a typical use of Lie algebra cohomology: an obstruction lives in a second cohomology group, and that group vanishes.