The killing form on a lie algebra is defined as(adsbygoogle = window.adsbygoogle || []).push({});

$$B(X,Y) = \text{Tr}ad_X \circ ad_Y$$

where ##ad_X: \mathfrak{g} \to \mathfrak{g}## is given by ##ad_X(Y) = [X,Y]##, where the latter is the lie bracket between X and Y in ##\mathfrak{g}##. Expressed in terms of components on a basis on ##\mathfrak{g}## we have

$$B_{ij} = c_{il}^{\ \ k} c_{jk}^{\ \ l}$$.

The killing form can serve as a biinvariant metric on the lie group G, and I've seen it stated several times that, if the group G is simple, then all other biinvariant metrics are proportional to the killing form. Especially the formula

$$B(X,Y) \sim \text{Tr}(\rho(X)\rho(Y))$$

where ##\rho## is a representation of ##\mathfrak{g}## is thrown a lot around.

So I wonder how this statement is proved?

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# Any biinvariant metric proportional to Killing metric

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