Diagonal-root SU(2) embedding in SO(5) ($⟨(x,y),(1,1{)}^{\vee }⟩=x+y$)

The long roots of a root system $\pm (1,1)$ and $\pm (1,-1)$ give the commuting $SU(2)$ factors of the standard $SO(4)$ subgroup. Choosing $(1,1)$ for the left factor pairs a weight $(x,y)$ with its coroot as $x+y$. The five-dimensional vector then gives two doublets and a singlet, whereas a short-axis root would give a triplet and two singlets.