Quaternions form Vector Space over Themselves

Theorem
The set of quaternions $\H$, with the operations of addition and multiplication, forms a vector space.

Proof
Let the set of set of quaternions be denoted $\struct {\C, +, \times}$.

From Quaternions form Skew Field, the algebraic structure $\struct {\H, +, \times}$ is a skew field.

By definition, a skew field is a division ring.

Also see

 * Properties of Quaternions