 Symmetric square of the sl3 representation of highest weight (2,1)

For the special linear Lie algebra $\mathfrak{sl}(3)$, with Dynkin labels indexing irreducible highest-weight representations,The summands have dimensions $60,24,15,15,6$, summing to $120=15\cdot 16/2$. The identity follows from the formal character of a weight module formula ${\chi }_{S^{2}W}(z)=({\chi }_W(z{)}^2+{\chi }_W(z^2))/2$ and the Weyl character formula.