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How can I find the Killing form of $\mathfrak{sp}_{2n}(\mathbb{C})$? I'm explicitly working with basis vectors in trying to compute $\operatorname{tr}(\operatorname{ad}(a)\operatorname{ad}(b))$ but it's becoming really messy. Is there another way?

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    For semisimple Lie algebras (like $\mathfrak{sp}_{2n}$), the Killing form is proportional to $(A,B) \mapsto tr(AB)$ (viewing elements $A$ and $B$ of the Lie algebra as matrices).2012-04-26

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