I am studying real representation from the book Representation theory by Fulton and Harris. Here I find an example which demonstrates the fact that if the character of a representation is real then the underlying representation need not be real. For instance the author take $G$ to be a finite non-abelian subgroup of $SU(2).$ To find this he identified $SU(2)$ with the unit quaternions. Now my question $:$ What are unit quaternions? What is difference of it with quaternion group? Any help will be warmly appreciated.
Thanks for your time.
Unit quaternions are the multiplicative group of the quaternions of norm $1$, where by norm we mean $$\lVert q\rVert=\sqrt{q_1^2+q_i^2+q_j^2+q_k^2}=\sqrt{q^*q}=\sqrt{qq^*}$$
and $(a+bi+cj+dk)^*=a-bi-cj-dk$ is the conjugate quaternion. The conjugate quaternion works in similar fashion to the complex conjugate, with the main thing to care about being that $(pq)^*=q^*p^*$. So, for instance, in the complex representation of the quaternion we have $(z_1+z_jj)^*=z_1^*-jz_j^*=z_1^*-z_jj$.
The quaternion group $Q_8$ usually indicates its finite subgroup $\{1,-1,i,-i,j,-j,k,-k\}$.