Show that the angular momentum Lie algebra (left[J_{i}, J_{j} ight]=i epsilon_{i j k} J_{k}) can be put

Show that the angular momentum Lie algebra \(\left[J_{i}, J_{j}\right]=i \epsilon_{i j k} J_{k}\) can be put in the form

\[\left[X_{1}, X_{2}\right]=X_{3} \quad\left[X_{2}, X_{3}\right]=X_{1} \quad\left[X_{3}, X_{1}\right]=X_{2},\]

by substituting \(J_{i} \rightarrow i X_{i}\), which is the form \(\left[X_{i}, X_{j}\right]=c_{i j}^{k} X_{k}\) described in Box 3.2 and assumed for Eq. (7.18).