= Traceless symmetric square of the defining even orthogonal representation
{title2=$\operatorname{Sym}^2(\mathbb C^{2n})=\mathbb C\oplus\operatorname{Sym}^2_0(\mathbb C^{2n})$}
For $n\geq2$, contraction with the invariant <symmetric bilinear form> splits off the trivial line in the <symmetric square> of the defining <special orthogonal group> <representation>. Its <null space> is the traceless <symmetric square>. It contains a <highest-weight vector> of <weight> $2e_1$, and the <Weyl dimension formula> gives $(n+1)(2n-1)$, exactly its <dimension>. <Complete reducibility of compact-group representations> therefore makes that <null space> <irreducible>. The entire <symmetric square> is reducible because of the invariant line.
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