SO5 to SO4 branching
= SO5 to SO4 branching
{c}
{title2=$SO(5)\downarrow SO(4)$}
Under the standard block-diagonal subgroup, the vector and adjoint representations branch as
$$
\mathbf5\to\mathbf4\oplus\mathbf1,
\qquad
\mathbf{10}\to\mathbf6\oplus\mathbf4.
$$
Using $\mathfrak{so}_4\cong\mathfrak{su}_2\oplus\mathfrak{su}_2$, these are $(\tfrac12,\tfrac12)\oplus(0,0)$ and $(1,0)\oplus(0,1)\oplus(\tfrac12,\tfrac12)$.