 Solution

For the defining representation, the standard coordinate vectors $e_1,e_2,e_3,e_4,e_5$ are weight vectors of weights ${\epsilon }_1,{\epsilon }_2,-{\epsilon }_1,-{\epsilon }_2,0$, respectively. Thus the weight diagram has four axial points and the origin, each of multiplicity one.

For the Adjoint representation, the eight nonzero weights of a representation are the roots from part (ii), each of multiplicity one, with the displayed $X_{\alpha }$ as basis. The zero weight space is the Cartan subalgebra, with basisThe Adjoint representation diagram therefore has the four axial points, four diagonal points, and the origin of multiplicity two. The multiplicities sum to ten, as they must.