= Solution
The <tensor-product weight diagram> is obtained by adding every ordered pair of defining <weights>. Each $\pm2L_i$ occurs once, while each of the four <weights> $\pm L_1\pm L_2$ occurs twice, from the two orders of the summands. Zero occurs four times, from the ordered pairs $(L_i,-L_i)$ and $(-L_i,L_i)$ for $i=1,2$. Hence
$$
\begin{array}{c|c}
\text{weight}&\text{multiplicity}\\ \hline
\pm2L_1,\ \pm2L_2&1\\
\pm L_1\pm L_2&2\\
0&4
\end{array}
$$
The <dimensions> sum to $4+8+4=16$, as required for $V\otimes V$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-2-sp4-tensor-weights.png]
{title=Tensor-square sp4 weights, including the double diagonal weights and four-dimensional zero space}
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