This came up in an exam on Lie algebras that I had today, and it's been bugging me. How do you prove that(adsbygoogle = window.adsbygoogle || []).push({});

[tex]B([X,Y],Z)=B(X,[Y,Z])[/tex]?

The best I've managed is writing

[tex]B([X,Y],Z)=\mathrm{Tr}(\mathrm{ad}([X,Y])\mathrm{ad}(Z))=\mathrm{Trace}([\mathrm{ad}(X),\mathrm{ad}(Y)]\mathrm{ad}(Z))[/tex]

but I have no idea where to go from there. Hints and/or a complete proof are both appreciated :)

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# Associativity of the Killing form

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