We can always normalize \(x = (a,b) \in C\) so that $$ a\overline{a} = b\overline{b} = 1 $$ Here \(a\) is a unit imaginary quaternion so \(a \in S^2 \), while \(b\) is a unit quaternion so \(b \in \mathrm{SU}(2) \cong S^3 \). Thus: $$ PC \cong \frac{S^2 \times \mathrm{SU}(2)}{(a,b) \sim (-a,-b)} $$ Proposition. There is a diffeomorphism $$ PC \cong \frac{S^2 \times \mathrm{SU}(2)}{(a,b) \sim (-a,b)} = \mathbb{R}\mathrm{P}^2 \times \mathrm{SU}(2) $$ sending \(\pm (a,b)\) to \( (\pm a,ab)\).