= Solution
The invariant <positive-definite bilinear form> makes every $\rho(g)$ orthogonal in an orthonormal real basis. Every element of $\mathrm{SU}(2)$ has the form $\begin{pmatrix}a&b\\-\overline b&\overline a\end{pmatrix}$ with $|a|^2+|b|^2=1$, so this group is homeomorphic to $S^3$ and connected. The <continuous function> $\det\rho(g)$ takes values in $\{-1,1\}$ and equals one at the identity. It is therefore identically one. Hence \b[$\boxed{\rho(\mathrm{SU}(2))\subseteq\mathrm{SO}(3)}$].
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