Cartan's Criterion - Cartan's Criterion For Solvability

Cartan's Criterion For Solvability

Cartan's criterion for solvability states:

A Lie subalgebra of endomorphisms of a finite dimensional vector space over a field of characteristic zero is solvable if and only if whenever

The fact that 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:

A finite-dimensional Lie algebra over a field of characteristic zero is solvable if and only if (where K is the Killing form).