Lie Algebra of su(2)
This is a simple exercise before jumping into the general theory of Lie algebra classification. The commutator relations in Lie algebra \mathfrak{su}(2) are \begin{equation}\Big[{\cal{J}}_a\,,{\cal{J}}_b\Big]=i\,\epsilon_{abc}\,{\cal{J}}_c\end{equation} for a,b,c=1,2,3. We construct an irreducible representation in the following steps: Pick any operator, say {\cal{J}}_3, and denote its eigenvector by |m\rangle: \begin{equation}{\cal{J}}_3\,|m\rangle = m\,|m\rangle\,.\tag{1}\end{equation} For the remaining {\cal{J}}_1 and {\cal{J}}_2, construct a set of new operators \begin{equation}{\cal{J}}_{\pm}\equiv {\cal{J}}_1 \pm i {\cal{J}}_2\,.\tag{2}\end{equation} The commutator relations become \begin{eqnarray}\Big[{\cal{J}}_3\,,{\cal{J}}_{\pm}\Big]=\pm {\cal{J}}_{\pm}\,,\quad \Big[{\cal{J}}_+\,,{\cal{J}}_-\Big]=2\,{\cal{J}}_3\,.\tag{3}\end{eqnarray} From the relation ${\cal{J}}_3\,{\cal{J}}_{\pm}\,|m\rangle = \Big[{\cal{J}}_3\,,{\cal{J}}_{\pm}\Big]\,|m\rangle + {\cal{J}}_{\p...