The Jacobi identity is a requirement of the commutation relations of a Lie algebra that ensures the corresponding Lie group is associative. For elements \(\hat{A}, \hat{B}, \hat{C}\) of a Lie algebra, the Jacobi identity is

\[\begin{equation*}[\hat{A},[\hat{B}, \hat{C}]]+[\hat{C},[\hat{A}, \hat{B}]]+[\hat{B},[\hat{C}, \hat{A}]]=0 \tag{8.144}\end{equation*}\]

In this expression, \([\hat{A}, \hat{B}]\) is called the Lie bracket of the Lie algebra. We have only studied Lie brackets that correspond to the familiar commutator, but it is possible to consider other definitions that satisfy the Jacobi identity.

(a) Show that the Jacobi identity is satisfied for the Lie algebra of threedimensional rotations, with Lie bracket

\[\begin{equation*}\left[\hat{L}_{i}, \hat{L}_{j}\right]=i \hbar \sum_{k=1}^{3} \epsilon_{i j k} \hat{L}_{k} \tag{8.145}\end{equation*}\]

(b) Show that the Jacobi identity is satisfied for any Lie algebra for which the Lie bracket is then just the commutator, that is, for

\[\begin{equation*}[\hat{A}, \hat{B}]=\hat{A} \hat{B}-\hat{B} \hat{A} \tag{8.146}\end{equation*}\]