SU(3) Lie algebra ($\mathfrak{su}(3)$)

The compact real Lie algebra $\mathfrak{su}(3)$ consists of traceless anti-Hermitian $3\times 3$ matrices. Physicists often specify Hermitian generators $t_a$ instead, so the genuine Lie algebra elements are $i{t}_a$. The compact real algebra and its complexification ${\mathfrak{sl}}_3(\mathbb{C})$ should be distinguished.

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.

// Target: lie-theory.bigb