It is claimed that $q = x{\bf i} + y{\bf j} + z{\bf k}$ has an one to one mapping to a vector $v \in \mathbb{R}^3$ where $v = x \hat i + y \hat j + z \hat k$
But ${\bf i}, {\bf j},{\bf k}$ are complex unit quaternion and $\hat i$ is a unit vector, so although they have similar notation, they are not the same quantity
What would be a mapping that takes a quaternion $q$ to a vector $v$?
Quaternions and $\mathbb{R}^4$ are isomorphes as vectorial spaces; you can apply the subspace of base {i, j, k} of $\mathbb{H}$ bijectively onto any subspace of $\mathbb{R}^4$ generated for any base {$b_1$,$b_2$,$b_3$} and in particular the base standard of your question.