Diagonal-root SU(2) embedding in SO(5)
= Diagonal-root SU(2) embedding in SO(5)
{title2=$\langle(x,y),(1,1)^\vee\rangle=x+y$}
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.