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.