Quaternions Subring of Complex Matrix Space

Theorem
The Ring of Quaternions is a subring of the matrix space $$\mathcal M_\C \left({2}\right)$$.

Proof
From Matrix Form of Quaternion it is clear that the quaternions $$\mathbb H$$ can be expressed in matrix form, as elements of $$\mathcal M_\C \left({2}\right)$$.

Thus $$\mathbb H \subseteq \mathcal M_\C \left({2}\right)$$.

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