Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 4 / d / Solution

The tensor product of the defining representation and its dual is the endomorphism representation. Splitting an endomorphism into its matrix trace and traceless part givesIn Dynkin labels, the two irreducible summands have highest weights $[0,0,0]$ and $[1,0,1]$.