Cartan's criterion

In mathematics, Cartan's criterion gives conditions for a Lie algebra in characteristic 0 to be solvable, which implies a related criterion for the Lie algebra to be semisimple. It is based on the notion of the Killing form, a symmetric bilinear form on ${\displaystyle {\mathfrak {g}}}$ defined by the formula

${\displaystyle B(u,v)=\operatorname {tr} (\operatorname {ad} (u)\operatorname {ad} (v)),}$

where tr denotes the trace of a linear operator. The criterion was introduced by Élie Cartan (1894).