In Jack Kuipers' book he says (p 114):
How can a quaternion, which lives in $\mathbb{R}^4$ operate on a vector, which lives in $\mathbb{R}^3$ ?
His answer:
A vector v $\in$ $\mathbb{R}^3$ can simply be treated as though it were a quaternion q $\in$ $\mathbb{R}^4$ whose real part is zero.
Then later we define that special property:
If p is a pure quaternion (ie $\Re({p}) =0 $): $ qpq* = w $
Where $\Re(w)=0$ as well.
Ok, so how does he get away with skipping over the fact that an $ \mathbb{R}^3 $ vector is real and a pure quaternion is purely imaginary?