= Solution
Write $V_r=\operatorname{Sym}^r\mathbb C^2$ for the irreducible <SU(2) representation> of dimension $r+1$, with <character of a representation> $z^r+z^{r-2}+\cdots+z^{-r}$. Here $V=V_3$, with weights $3,1,-1,-3$. Squaring its <character of a representation> and successively removing the highest-weight irreducible characters gives
$$
\boxed{V\otimes V\cong V_6\oplus V_4\oplus V_2\oplus V_0.}
$$
The dimensions $7+5+3+1=16$ check the decomposition; complete reducibility follows from the invariant inner product.
For the <exterior square>, add distinct pairs of weights. Its <character of a representation> is $z^4+z^2+2+z^{-2}+z^{-4}=\chi_{V_4}+\chi_{V_0}$. Equivalently it is $(\chi_V(z)^2-\chi_V(z^2))/2$. Hence $\boxed{\Lambda^2V\cong V_4\oplus V_0}$, of dimensions five and one. The complementary <symmetric square> is $V_6\oplus V_2$.
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