Quaternions Subring of Complex Matrix Space

Theorem
The Ring of Quaternions is a subring of the matrix space $\map {\MM_\C} 2$.

Proof
From Matrix Form of Quaternion it is clear that the quaternions $\H$ can be expressed in matrix form, as elements of $\map {\MM_\C} 2$.

Thus $\H \subseteq \map {\MM_\C} 2$.

As the quaternions form a ring, the result follows by definition of subring.