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Cartan's criterion

Cartan's criterion (by Élie Cartan) gives conditions for a finite-dimensional Lie algebra of characteristic zero to be solvable or semisimple. It is expressed in terms of the Killing form, the symmetric bilinear form on g {\displaystyle {\mathfrak {g}}} defined by

where tr denotes the trace of a linear operator. Namely: a Lie algebra g {\displaystyle {\mathfrak {g}}} is semisimple if and only if the Killing form is nondegenerate. A Lie algebra g {\displaystyle {\mathfrak {g}}} is solvable if and only if K (g,) = 0. {\displaystyle K({\mathfrak {g}},)=0.}