Show that the four successive infinitesimal rotations ( $\varepsilon$ about the 1-axis, followed by $\eta$ about the 2 -axis, then $-\varepsilon$ about the 1 -axis, and finally $-\eta$ about the 2 -axis) are equivalent to the second-order rotation $\varepsilon \eta$ about the 3-axis. Hence, show that the generators satisfy
$$
\left[J_1, J_2\right]=i J_3 \text {. }
$$
Nonlinear functions of the generators which commute with all the generators are called invariants or Casimir operators. For the rotation group,
$$
J^2=J_1^2+J_2^2+J_3^2
$$
is the only Casimir operator,
$$
\left[J^2, J_i\right]=0 \quad \text { with } i=1,2,3 .
$$
It follows that we can construct simultaneous eigenstates $|j m\rangle$ of $J^2$ and one of the generators, say $J_3$. Using only (2.13), it is possible to show that
$$
\begin{aligned}
J^2|j m\rangle & =j(j+1)|j m\rangle \\
J_3|j m\rangle & =m|j m\rangle
\end{aligned}
$$
with $m=-j,-j+1, \ldots, j$, and where $j$ can take one of the values $0, \frac{1}{2}, 1, \frac{3}{2}, \ldots$