= Solution
For the <Adjoint representation>, the <root-space decomposition> gives the eight nonzero <weights>
$$
\pm2L_1,\quad\pm2L_2,\quad\pm L_1\pm L_2,
$$
each of multiplicity one, and <weight> zero of multiplicity two from the <Cartan subalgebra>. Thus its <weight diagram> is the <root> diagram with a double zero <weight>, and its total <dimension> is $8+2=10$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-2-sp4-adjoint-weights.png]
{title=Adjoint sp4 weights: the eight roots and the zero weight of multiplicity two}
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