Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 3 / d / Solution

Since $-\theta T_1=i\theta {\sigma }_1/2$, the same disentangling identity applies in every Lie algebra representation. The rightmost $E_{+}$ kills $|\Lambda ⟩$, the leftmost $E_{-}$ kills $⟨\Lambda |$, and the middle factor acts by its highest weight. Therefore