= Solution
Take the <simple roots> $L_1-L_2$ and $L_2-L_3$. A weight represented by $(k_1,k_2,k_3)$ is in the <Closed dominant Weyl chamber> exactly when $k_1\geq k_2\geq k_3$. Among nonnegative triples of total four, the possibilities are $(4,0,0),(3,1,0),(2,2,0),(2,1,1)$. Using $L_1+L_2+L_3=0$, their explicit <weight space> bases are
$$
\begin{array}{c|l}
\text{weight}&\text{basis}\\\hline
4L_1&s_{11}\otimes s_{11}\\
2L_1-L_3&s_{11}\otimes s_{12},\ s_{12}\otimes s_{11}\\
-2L_3&s_{11}\otimes s_{22},\ s_{22}\otimes s_{11},\ s_{12}\otimes s_{12}\\
L_1&s_{11}\otimes s_{23},\ s_{23}\otimes s_{11},\ s_{12}\otimes s_{13},\ s_{13}\otimes s_{12}
\end{array}
$$
Each listed tensor has the required summed weight, and the monomial basis from part (i) shows linear independence and exhausts all possible tensors of that weight. Thus these are all the requested bases.
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