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The wikipedia page on Killing form reports a nice table with an explicit calculation of the Killing form for many classical real Lie algebras, in terms of traces of the defining representation.

I am aware that there is a closely related question here, but I am looking for an "academic" reference (book or peer-reviewed article) for this table.

francesco
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  • I don't know about a specific reference with all those Killing forms being computed explicitly, but the main idea is to use Schur's Lemma in the case that your Lie algebra is simple. When it is semisimple, one decomposes it into simple ideals and then the Killing form is the orthogonal direct sum of the Killing forms in each factor. – Johnny El Curvas Jul 25 '22 at 10:50
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    You could cite the book by Erdmann and Wildon, where it is in the exercises. – Dietrich Burde Jul 25 '22 at 11:08

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