# Physics-y resource request on Killing-Cartan forms

Most books that treat nonabelian gauge theory do not contain detailed discussion on Killing-Cartan forms, they'll usually just say that in $$\text{SU}(N)$$ Yang-Mills theory, one can choose generators $$T_A$$ such that $$\text{Tr}(T_AT_B)=\frac{1}{2}\delta_{AB},$$ implying that the inner product in the Lie algebra is $$\langle T_A, T_B\rangle=2\text{Tr}(T_A T_B).$$

Even on the mathematical side, I mainly studied Lie groups from differential geometry texts that often completely ignored the Killing-Cartan form, or just barely mentioned it.

In short, I am looking for mainly physics-oriented texts that treat it in detail, including but not limited to

• deriving necessary and sufficient conditions on the Lie group/algebra such that the Killing form is nondegenerate and definite (emphasis is on deriving, I can look up the conditions themselves in wikipedia)

• various conventions and expressions relating it to the trace operation in matrix representations.

Whilst, as I have stated beforehand, I do not mind, in fact I even prefer physics-minded treatments, I don't even mind if the reference is working solely with matrix groups, I would like all proofs/derivations to be rigorous, rather than handwavy. I can get all facts from texts I already have, I am solely looking for proofs.

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