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and are orthogonal subspaces with respect to the Killing form.
* The orthogonal complement with respect to of an ideal is again an ideal.
* If a given Lie algebra is a direct sum of its ideals , then the Killing form of is the direct sum of the Killing forms of the individual summands.

Matrix elements

Given a basis of the Lie algebra , the matrix elements of the Killing form are given by

:

Here

:

in Einstein summation notation,

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