The weights of $V_2$ are $z^2,1,z^{-2}$. Degree-three monomials in these weights show that the character of the <symmetric power> $S^3V_2$ has weights
$$
z^6+z^4+2z^2+2+2z^{-2}+z^{-4}+z^{-6}
=\chi_{V_6}+\chi_{V_2}.
$$
Characters determine finite-dimensional $SU(2)$ representations, hence
$$
\boxed{S^3V_2\cong V_6\oplus V_2}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 19I / a / iii / Solution

{title2=$S(V)$}
{wiki}

The symmetric algebra is the quotient of $T(V)$ by the ideal generated by $v\otimes w-w\otimes v$. It is graded by <symmetric powers>:
$$
S(V)=\bigoplus_{n\geq0}S^nV.
$$

Symmetric algebra

{title2=$S^nV$}
{wiki}

The nth symmetric power $S^nV$ is the quotient of $V^{\otimes n}$ that identifies tensors differing by a permutation of their factors. Equivalently, over characteristic zero it is the subspace of symmetric tensors.


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 Incoming links: Symmetric power