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.