The Killing form is an inner product on a finite dimensional Lie algebra defined by
(1)
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in the adjoint representation, where is the adjoint representation of . (1) is adjoint-invariant in the sense that
(2)
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When is a semisimple Lie algebra, the Killing form is nondegenerate.
For example, the special linear Lie algebra has three basis vectors , where :
(3)
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(4)
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(5)
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The other brackets are given by and . In the adjoint representation, with the ordered basis , these elements are represented by
(6)
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(7)
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(8)
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and so where
(9)
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