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:
 * $$\mathbf i \mathbf j = k, \mathbf j \mathbf i = -k$$

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