- Cartan's criterion
**Cartan's criterion**is an important mathematical theorem in the foundations ofLie algebra theory that gives conditions for a Lie agebra to be nilpotent, solvable, or semisimple. It is based on the notion of theKilling form , asymmetric bilinear form on $mathfrak\{g\}$ defined by the formula: $K(u,v)=operatorname\{tr\}(operatorname\{ad\}(u)operatorname\{ad\}(v)),$where tr denotes the trace of a linear operator. The criterion is named afterÉlie Cartan .**Formulation**Cartan's criterion states: : "A finite-dimensional Lie algebra $mathfrak\{g\}$ over a field of

characteristic zero is semisimple if and only if the Killing form isnondegenerate . A Lie algebra $mathfrak\{g\}$ is solvable if and only if $K(mathfrak\{g\},\; [mathfrak\{g\},mathfrak\{g\}]\; )=0.$"More generally, a finite-dimensional Lie algebra $mathfrak\{g\}$ is reductive if and only if it admits a nondegenerate

invariant bilinear form .**References***

Jean-Pierre Serre , "Lie algebras and Lie groups." 1964 lectures given at Harvard University. Second edition. Lecture Notes in Mathematics, 1500. Springer-Verlag, Berlin, 1992. viii+168 pp. ISBN 3-540-55008-9**See also***

Modular Lie algebra

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