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