Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 302 / 1 / iv

The positive roots generated by the two simple roots areThe Adjoint representation of a Lie algebra has one weight space for each positive and negative root and a rank-two zero-weight space. Its nonzero weights are thereforetogether with their negatives. Thus the diagram consists of a hexagon of six long roots, a hexagon of six short roots, and the origin with multiplicity two. The representation has dimension $12+2=14$.