The Killing form of a Lie algebra provides the definition of normalization of operators in a particular representation of the Lie algebra. For representation \(R\) of \(\mathfrak{s u}(2)\), the Killing form is

\[\begin{equation*}\operatorname{tr}\left[\hat{L}_{i}^{(R)} \hat{L}_{j}^{(R)}\right]=k_{R} \delta_{i j} \tag{8.147}\end{equation*}\]

where \(\operatorname{tr}\) denotes the trace (i.e., the sum of diagonal elements) and \(\hat{L}_{i}^{(R)}\) is an element in the representation \(R\) of the Lie algebra. The quantity \(k_{R}\) depends on the representation. For a representation \(R\) of dimension \(D\) and \(\operatorname{Casimir} C_{R}\), where

\[\begin{equation*}C_{R} \mathbb{I}_{D}=\left(\hat{L}_{x}^{(R)}\right)^{2}+\left(\hat{L}_{y}^{(R)}\right)^{2}+\left(\hat{L}_{z}^{(R)}\right)^{2} \tag{8.148}\end{equation*}\]

and \(\mathbb{I}_{D}\) is the \(D \times D\) identity matrix, express \(k_{R}\) in terms of \(D\) and \(C_{R}\). What is \(k_{R}\) for a representation of spin \(\ell\) ?