= Solution
Let $e_1,e_2,e_3$ be the <basis> of the <defining representation> and $y_1,y_2,y_3$ its dual. On a diagonal element of the <Cartan subalgebra> their <weights> are $L_i$ and $-L_i$, with $L_1+L_2+L_3=0$. Use the <tensor product> and <symmetric square> <basis>
$$
T_{i;jk}=e_i\otimes y_jy_k,\qquad 1\leq i\leq3,\quad1\leq j\leq k\leq3.
$$
Its <weight> is $L_i-L_j-L_k$. For the three repeated <weights> put
$$
A_j=T_{j;jj},\qquad B_{i;j}=e_i\otimes y_i y_j\quad(i\ne j).
$$
If $\{i,l\}=\{1,2,3\}\setminus\{j\}$, the complete <weight-space decomposition> is
$$
\begin{array}{c|c|c}
\text{weight}&\text{basis}&\text{multiplicity}\\ \hline
L_i-2L_j\ (i\ne j)&T_{i;jj}&1\\
2L_i&T_{i;jk},\quad\{i,j,k\}=\{1,2,3\},\ j<k&1\\
-L_j&A_j,\ B_{i;j},\ B_{l;j}&3
\end{array}
$$
There are six spaces of the first type, three of the second, and three of the third. Their <dimensions> sum to $6+3+9=18$. The <weight diagram> below labels every space with all of its <basis vectors>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-2-sl3-tensor-weights.png]
{title=All weight spaces of the sl3 tensor module, with each tensor-basis vector shown at its weight}
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