In reading a text on quantum mechanics and symmetry*, I have encountered a statement of Cartan's "criterion for semisimplicity" and a proof. The statement reads: a Lie algebra is semisimple if and only if det g ne 0. The g is an GxG matrix, where G is the order of the generators of unitary transformations of vectors in a Hilbert space. "g" is called a metric tensor and is built from the "structure constants" of the generators. It is also called the "Killing form".
Starting with the hypothesis that the Lie algebra has an Abelian sub-algebra, the author shows clearly that the det g = 0: the "if" above.
The author then proceeds with his proof that if the det g = 0, the Lie algebra is not semisimple; i.e. it has an Abelian subalgebra, the "only if" part of the "criterion" above.
I do not find the proof convincing. Does anyone know if the "only if" part of the above stated criterion is correctly stated? A copy of the proof or a reference would be much appreciated.
* Quantum Mechanics Symmetries (2nd Edition) by Greiner & Mueller.