 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 $\left(\begin{array}{cc}a& b\\ -\overline{b}& \overline{a}\end{array}\right)$ 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 $\boxed{\rho (\mathrm{SU}(2))\subseteq \mathrm{SO}(3)}$.