so5 Lie algebra (${\mathfrak{so}}_5$)

The Lie algebra ${\mathfrak{so}}_5$ has dimension ten and root system $B_2$.

Under the standard block-diagonal subgroup, the vector and adjoint representations branch asUsing ${\mathfrak{so}}_4\cong {\mathfrak{su}}_2\oplus {\mathfrak{su}}_2$, these are $(\frac{1}{2},\frac{1}{2})\oplus (0,0)$ and $(1,0)\oplus (0,1)\oplus (\frac{1}{2},\frac{1}{2})$.

Identify ${\mathbb{R}}^4$ with the quaternions. Unit quaternions act by left multiplication, embedding the SU(2) group in $SO(4)$. Under this subgroup, the complexified vector representation branches as $4\to 22$, and the Adjoint representation as $6\to 3\oplus 31$. Right multiplication supplies the commuting second factor.

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.