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).
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