Let be a finite dimensional central simple -algebra. We proved in this theorem that is a group homomorphism. The image of is an abelian group because it lies in So if then Therefore the commutator subgroup of is contained in and so the following definition makes sense.

**Definition**. Let be a finite dimensional central simple algebra. The **reduced Whitehead group** of is the factor group

**Remark**. Let be a finite dimensional central division -algebra of degree By the theorem in this post, for every there exists such that So if then Therefore for all

**Example 1**. if is a field and either or and

*Proof*. Let Then and

We proved in here that Thus

**Example 2**. where is the division algebra of quaternions over

*Proof*. Let and suppose that We need to prove that is in the commutator subgroup of We are going to prove a stronger result, i.e. for some If then and If then we can choose and if we can choose So we may assume that We also have (you may either directly check this or use this fact that every element of a finite dimensional central simple algebra satisfies its reduced characteristic polynomial). Thus Note that since and we have So for some Let

So, since the real part of is the real part of is zero and hence

Since the center of is and there exists such that Therefore

Now, by we have and So we can write and Let Then gives us

We also have by and Thus, by again,

To be continued …