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 :)

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Associativity of the Killing form

Loading...

Similar Threads - Associativity Killing form | Date |
---|---|

I Is a commutative A-algebra algebraic over A associative? | Feb 6, 2018 |

I Index (killing form ?) in a reducible representation | Jul 25, 2017 |

B Associativity of Matrix multiplication | Jun 9, 2017 |

B Mistake when explaining associativity of vector addition | Jul 21, 2016 |

I K-algebra/associative algebra - equivalence of definitions | Jun 13, 2016 |

**Physics Forums - The Fusion of Science and Community**