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Consider $sl_n\mathbb{C}$ as aLie-algebra, and choose h the CSA formed by diagonal matrixes. I can i demonstrate that the Cartan-Killing form in $sl_n\mathbb{C}$ is $=2n \sum_{i=1}^n a_ib_i$?

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    Just look at http://math.stackexchange.com/questions/135567/computation-of-the-killing-form-of-mathfrakgl-m/1248343#1248343 and put $tr(A)=tr(B)=0$ condition at the result.2015-04-23

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