Given a semisimple and compact Lie algebra $L$ generated by a certain number of elements $\{t_i\}$ the Cartan-Killing metric is defined as: $$k_{ij} \equiv tr(ad_{t_i}ad_{t_j})$$ where $ad_{t_i}$ is the adjoint representation of $t_i$ and the metric is nonsingular and negative definite.
In chapter 15 of the Penskin (An Introduction To Quantum Field Theory) it is written that for every irreducible representation $t^r_i$ of $t_i$ we have that: $$tr(t^r_it^r_j) = c_rtr(ad_{t_i}ad_{t_j})$$ and thus we can compute the Cartan Killing metric in any irreducible representation up to a multiplicative factor $c_r$ which depends on the representation.
I tried to look for a proof of this equality but couldn't find anything. Could someone tell me where it comes from even without giving the complete proof?
