= Spinor tensor square for so5
{title2=$\mathbf4\otimes\mathbf4=\mathbf1\oplus\mathbf5\oplus\mathbf{10}$}
For the <B2 root system> with long <simple root> first, the four-dimensional <spinor representation> has <highest weight> $(0,1)$. Its <symmetric square> has <highest weight> $(0,2)$ and <dimension> ten, so it is the <Adjoint representation>. The <exterior square> has the four nonzero defining <weights> and two zero <weights>; it splits into the five-dimensional <vector representation> of <highest weight> $(1,0)$ and a <trivial representation>. Thus the <tensor square> decomposes as displayed. The invariant alternating <bilinear form> reflects the <pseudoreal representation> structure. The <spinor representation> is a <group representation> of the <Spin group> $\operatorname{Spin}(5)$, while its <tensor square> descends to $SO(5)$.
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