| 1. | The Cartan subalgebra inherits an inner product from the Killing form on.
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| 2. | A natural inner product on \ mathfrak g is given by the Killing form.
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| 3. | The dual cone with respect to the Killing form is the maximal invariant convex cone.
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| 4. | The converse is false : there are non-nilpotent Lie algebras whose Killing form vanishes.
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| 5. | Derivations on \ mathfrak { h } are skew-adjoint for the inner product given by minus the Killing form.
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| 6. | The second one is the compact real form and its Killing form is negative definite, i . e . has signature.
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| 7. | The Killing form and the Casimir invariant also have a particularly simple form, when written in terms of the structure constants.
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| 8. | The Killing form is negative definite on the + 1 eigenspace of ? and positive definite on the " 1 eigenspace.
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| 9. | The real forms of a given complex semisimple Lie algebra are frequently labeled by the positive index of inertia of their Killing form.
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| 10. | By Cartan's criterion, the Killing form is nondegenerate, and can be diagonalized in a suitable basis with the diagonal entries.
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