Cartan's criterion
= Cartan's criterion
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For a finite-dimensional <Lie algebra> over a field of characteristic zero, <Cartan's criterion> characterizes a <solvable Lie algebra> by vanishing of its <Killing form> on $\mathfrak g\times[\mathfrak g,\mathfrak g]$, and a <semisimple Lie algebra> by nondegeneracy of that form. The <Cartan solvability criterion> and <Cartan criterion for semisimplicity> state the two distinct tests.