= sl2 tensor product of cubic and quadratic symmetric powers
{title2=$S^3V\otimes S^2V\cong\Gamma_5\oplus\Gamma_3\oplus\Gamma_1$}
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.
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