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Is there a more intelligent way to tackle exercise 7 of paragraph 5 of Humphreys (Introduction to Lie Algebras and Representation Theory)?

Exercise 7: Relative to the standard basis of $L = \mathfrak{sl}(3,F)$, compute the determinant of the Killing form.

It seems to me a lot of work to write down $\text{ad } x$ with respect to the standard basis of $L$ (with $x$ running through the standard basis of $L = \mathfrak{sl}(3,F)$) and then calculate the matrix corresponding to the Killing Form.

Thanks in advance.

AYK
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    Could you write out the exercise? – anomaly Mar 13 '15 at 13:01
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    I have voted to close as "missing context". If you update the question to actually include the exercise I will remove the vote again (assuming you ping me so I remember). – Tobias Kildetoft Mar 13 '15 at 13:14
  • Ok thanks, I have added the exercise. – AYK Mar 13 '15 at 18:12
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    Well, the problem with this question is that you seem to think that that computation is a lot of work, when it isn't! – Mariano Suárez-Álvarez Mar 13 '15 at 18:41
  • Could you explain why not? I mean using the fact that some of the matrices (of the standard basis) are the transposed of each other I can reduce some calculations. But still there remains a lot to calculate. – AYK Mar 14 '15 at 02:14

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