Quaternions form Skew Field

Theorem
The set $\mathbb H$ of quaternions forms a skew field under the operations of addition and multiplication.

Proof
From Ring of Quaternions we have that $\mathbb H$ forms a ring.

From Multiplicative Identity for Quaternions‎ we have that $\mathbf 1$ is the identity for quaternion multiplication.

From Multiplicative Inverse of Quaternion we have that every element of $\mathbb H$ except $\mathbf 0$ has an inverse under quaternion multiplication.

So $\mathbb H \setminus \left\{{0}\right\} = \mathbb H^*$ is a group.

Hence $\mathbb H$ forms a division ring.

But quaternion multiplication is specifically not commutative, for example:
 * $\mathbf i \mathbf j = \mathbf k, \mathbf j \mathbf i = -\mathbf k$

So $\mathbb H$ forms a skew field under addition and multiplication.