Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 302 / 4 / f / Solution

Under $\mathfrak{sl}(2{)}_{\alpha }$, a root $x\alpha +y\beta$ has weight $2x-y$. Counting the twelve root spaces and the two-dimensional Cartan subalgebra gives multiplicities one at weights $\pm 2$, four at $\pm 1$, and four at zero. Therefore

Under $\mathfrak{sl}(2{)}_{\beta }$, the weight is $-3x+2y$. The multiplicities are two at $\pm 3$, one at $\pm 2$, two at $\pm 1$, and four at zero. HenceThe dimensions are respectively $3+4\cdot 2+3=14$ and $2\cdot 4+3+3=14$, verifying both direct-sum decompositions of the Adjoint representation.