= Solution
Write $L(n)$ for the <irreducible representation> of $\mathfrak{sl}_2$ with highest weight $n$, dimension $n+1$ and weights $n,n-2,\ldots,-n$. The <weight diagram> below records multiplicities $1,2,2,2,1$ at weights $4,2,0,-2,-4$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-1-sl2-weights.png]
{title=Weights and irreducible summands of the defining sl2 module tensor its cubic symmetric power}
The unique top weight four forces one copy of $L(4)$, by <complete reducibility of semisimple Lie algebra representations>. Removing its five weights, each with multiplicity one, leaves weights $2,0,-2$ with multiplicity one, exactly the character of $L(2)$. Hence
$$
\boxed{U\cong L(4)\oplus L(2),\qquad\dim U=5+3.}
$$
The two lower rows draw their separate <weight diagrams>. The explicit invariant copies constructed next establish the <Clebsch-Gordan splitting of a defining module and its cubic symmetric power> within $U$ itself.
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