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

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

Cartan's criterion for solvability states:

The fact that ${\displaystyle Tr(ab)=0}$ in the solvable case follows immediately from Lie's theorem that solvable Lie algebras in characteristic 0 can be put in upper triangular form.

Applying Cartan's criterion to the adjoint representation gives:

Cartan's criterion for semisimplicity states:

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