Recall that the real quaternion algebra, as a vector space over has a basis and multiplication of two elements in is done using distributive law and the relations It is easy to see that is a division algebra and its center is So, by the lemma in this post, This isomorphism is also a result of this fact that is a maximal subfield of (see corollary 3 in this post!). In the following example, we are going to find the reduced characteristic polynomial of an element of by giving a -algebra isomorphism explicitely. We will learn later how to find the reduced characteristic polynomial of an element of any finite dimensional central division algebra.

**Example. **If , then

**Solution**. We define as follows: for every and we define

Then of course we extend linearly to all elements of See that is a -algebra homomorphism. Since is simple, is injective. Therefore is an isomorphism because It is easy now to find :

The rest of the proof is straightforward.

In the above example, the coefficients of are all in This is not an accident. We have already proved that the reduced characteristic polynomial of an element of a finite dimensional central simple -algebra is always in (see the theorem in this post!).