This map takes the 2-sphere of radius inside the purely imaginary quaternions to \{s\in S^3 \subset \bold{H}: \operatorname{Re}(s) = \cos(R)\} , a 2-sphere of radius \sin(R) when R\not\equiv 0\pmod{2\pi}.
Let V be a finite dimensional real vector space and view it as an additive Lie group.