= Tensor-square decomposition of the defining so5 representation
{c}
{title2=$V(\omega_2)^{\otimes2}=V(0)\oplus V(2\omega_1)\oplus V(2\omega_2)$}
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
$$
V(\omega_2)\otimes V(\omega_2)
\cong V(0)\oplus V(2\omega_1)\oplus V(2\omega_2).
$$
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