SO5 to SO4 branching ($SO(5)\downarrow SO(4)$)

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 $(1,0)\oplus (0,1)\oplus (\frac{1}{2},\frac{1}{2})$.