 Root SU(2) subalgebras of SU(3)

Each root $\alpha$ of the SU(3) Lie algebra defines an SU(2) Lie algebra generated by $J_3=\alpha \cdot H$ and $J_{\pm }=\sqrt{2}{E}_{\pm \alpha }$ in the root-length-one convention. They obey $[J_3,J_{\pm }]=\pm J_{\pm }$ and $[J_{+},J_{-}]=2J_3$. These subalgebras explain root strings, Weyl group reflections and the integer Dynkin labels of highest weights.

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