Inmathematics,Cartan's criterion gives conditions for aLie algebra in characteristic 0 to besolvable, which implies a related criterion for the Lie algebra to besemisimple. It is based on the notion of theKilling form, asymmetric bilinear form on${\displaystyle \mathfrak{g}}$ defined by the formula

${\displaystyle \kappa (u,v)=\mathrm{tr}(\mathrm{ad}(u)\mathrm{ad}(v)),}$

where tr denotes thetrace of a linear operator. The criterion was introduced byÉlie Cartan (1894).^[1]

Cartan's criterion for solvability states:

A Lie subalgebra${\displaystyle \mathfrak{g}}$ of endomorphisms of a finite-dimensional vector space over afield ofcharacteristic zero is solvable if and only if${\displaystyle \mathrm{tr}(ab)=0}$ whenever${\displaystyle a\in \mathfrak{g},b\in [\mathfrak{g},\mathfrak{g}].}$

The fact that${\displaystyle \mathrm{tr}(ab)=0}$ in the solvable case follows fromLie's theorem that puts${\displaystyle \mathfrak{g}}$ in the upper triangular form over the algebraic closure of the ground field (the trace can be computed after extending the ground field). The converse can be deduced from thenilpotency criterion based on theJordan–Chevalley decomposition, as explained there.

Applying Cartan's criterion to the adjoint representation gives:

A finite-dimensional Lie algebra${\displaystyle \mathfrak{g}}$ over afield ofcharacteristic zero is solvable if and only if${\displaystyle \kappa (\mathfrak{g},[\mathfrak{g},\mathfrak{g}])=0}$ (where${\displaystyle \kappa }$ is the Killing form).

Cartan's criterion for semisimplicity states:

A finite-dimensional Lie algebra${\displaystyle \mathfrak{g}}$ over afield ofcharacteristic zero is semisimple if and only if the Killing form isnon-degenerate.

Jean Dieudonné (1953) gave a very short proof that if a finite-dimensional Lie algebra (in any characteristic) has anon-degenerate invariant bilinear form and no non-zero abelian ideals, and in particular if its Killing form is non-degenerate, then it is a sum of simple Lie algebras.

Conversely, it follows easily from Cartan's criterion for solvability that a semisimple algebra (in characteristic 0) has a non-degenerate Killing form.

Cartan's criteria fail in characteristic${\displaystyle p>0}$; for example:

• the Lie algebra${\displaystyle {\mathrm{SL}}_p(k)}$ is simple ifk has characteristic not 2 and has vanishing Killing form, though it does have a nonzero invariant bilinear form given by${\displaystyle (a,b)=\mathrm{tr}(ab)}$.
• the Lie algebra with basis${\displaystyle a_n}$ for${\displaystyle n\in \mathbb{Z}/p\mathbb{Z}}$ and bracket [a_i,a_j] = (i−j)a_i+j is simple for${\displaystyle p>2}$ but has no nonzero invariant bilinear form.
• Ifk has characteristic 2 then thesemidirect product gl₂(k).k² is a solvable Lie algebra, but the Killing form is not identically zero on its derived algebra sl₂(k).k².

If a finite-dimensional Lie algebra is nilpotent, then the Killing form is identically zero (and more generally the Killing form vanishes on any nilpotent ideal). The converse is false: there are non-nilpotent Lie algebras whose Killing form vanishes. An example is given by the semidirect product of an abelian Lie algebraV with a 1-dimensional Lie algebra acting onV as an endomorphismb such thatb is not nilpotent and Tr(b²)=0.

In characteristic 0, every reductive Lie algebra (one that is a sum of abelian and simple Lie algebras) has a non-degenerate invariant symmetric bilinear form. However the converse is false: a Lie algebra with a non-degenerate invariant symmetric bilinear form need not be a sum of simple and abelian Lie algebras. A typical counterexample isG =L[t]/tⁿL[t] wheren>1,L is a simple complex Lie algebra with a bilinear form (,), and the bilinear form onG is given by taking the coefficient oftⁿ⁻¹ of theC[t]-valued bilinear form onG induced by the form onL. The bilinear form is non-degenerate, but the Lie algebra is not a sum of simple and abelian Lie algebras.

• Cartan, Élie (1894),Sur la structure des groupes de transformations finis et continus, Thesis, Nony
• Dieudonné, Jean (1953), "On semi-simple Lie algebras",Proceedings of the American Mathematical Society,4 (6):931–932,doi:10.2307/2031832,ISSN 0002-9939,JSTOR 2031832,MR 0059262
• Serre, Jean-Pierre (2006) [1964],Lie algebras and Lie groups, Lecture Notes in Mathematics, vol. 1500, Berlin, New York:Springer-Verlag,doi:10.1007/978-3-540-70634-2,ISBN 978-3-540-55008-2,MR 2179691

• Modular Lie algebra

• Lie algebras