Tensor-square decomposition of the defining so5 representation ($V({\omega }_2{)}^{\otimes 2}=V(0)\oplus V(2{\omega }_1)\oplus V(2{\omega }_2)$)

For the five-dimensional defining representation of ${\mathfrak{so}}_5$, the symmetric square is the sum of its invariant trace and the traceless representation $V(2{\omega }_2)$, while the exterior square is the Adjoint representation $V(2{\omega }_1)$. Hence