sl2 tensor product of cubic and quadratic symmetric powers ($S^{3}V\otimes S^{2}V\cong {\Gamma }_5\oplus {\Gamma }_3\oplus {\Gamma }_1$)

For the defining two-dimensional sl2 Lie algebra module with coordinates $x,y$, put $u_{ij}=x^{3-i}{y}^i\otimes x^{2-j}{y}^j$. Its three highest-weight vectors are $u_{00}$, $u_{10}-u_{01}$ and $u_{20}-2u_{11}+u_{02}$, of weights five, three and one. Lowering the last once gives $u_{30}-2u_{21}+u_{12}$, an explicit second vector of its two-dimensional irreducible summand.