For a semisimple compact Lie algebra \begin{eqnarray} [T_a,T_b]=if_{ab}^cT_c, \end{eqnarray} the Killing form component takes the form as \begin{eqnarray} \kappa_{ab}=-\text{Tr}\left[T_a^AT_b^A\right]=-\left(T_a^A\right)_k^l\left(T_b^A\right)_l^k \end{eqnarray} where ''$A$'' means the adjoint representation \begin{eqnarray} \left(T_a^A\right)_b^c\equiv-if_{ab}^c. \end{eqnarray} In many textbooks, it is directly given that, for any other representation ''$R$'', an analogue of $\kappa_{ab}$ defined by \begin{eqnarray} \kappa_{ab}^R\equiv-\text{Tr}\left[T_a^RT_b^R\right]\propto\kappa_{ab}. \end{eqnarray} I cannot obtain the proportionality above.
Could someone help me reach it and understand it?