= Solution
Use $h=\operatorname{diag}(1,-1)$, $e=E_{12}$ and $f=E_{21}$ in the <sl2 Lie algebra>, so $hx=x$, $hy=-y$, $ey=x$ and $fx=y$. On the <symmetric power>, a monomial $x^{3-k}y^k$ has weight $3-2k$. The <tensor product of Lie algebra representations> adds weights, because $h$ acts as $h\otimes I+I\otimes h$.
Put $A_k=x\otimes x^{3-k}y^k$ and $B_k=y\otimes x^{3-k}y^k$ for $0\le k\le3$. These eight vectors are a <basis> of $U$; their weights are $4-2k$ and $2-2k$. Thus the <weight-space decomposition> and a <weight basis> for each summand are
$$
\begin{array}{c|c|c}
\text{weight}&\text{basis of its weight space}&\text{multiplicity}\\\hline
4&A_0&1\\
2&A_1,\ B_0&2\\
0&A_2,\ B_1&2\\
-2&A_3,\ B_2&2\\
-4&B_3&1
\end{array}
$$
In particular $\boxed{U=U_4\oplus U_2\oplus U_0\oplus U_{-2}\oplus U_{-4}}$, with dimensions summing to eight. Here the subscripts are the eigenvalues of the chosen Cartan element $h$.
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