= Tensor-square decomposition of the defining even orthogonal representation
{title2=$V\otimes V=L(2\varepsilon_1)\oplus L(\varepsilon_1+\varepsilon_2)\oplus L(0)$}
For $V=\mathbb C^{2n}$, $n\geq3$, the <symmetric square> splits into an invariant line and its irreducible traceless part of highest weight $2\varepsilon_1$. The <exterior square of the defining orthogonal representation> is the adjoint module of highest weight $\varepsilon_1+\varepsilon_2$. For $n=2$, the exterior square splits instead into highest weights $\varepsilon_1-\varepsilon_2$ and $\varepsilon_1+\varepsilon_2$. This exception reflects that <so4 Lie algebra> is semisimple but not simple.
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