Exercise 8, § 2

The quaternionic group $Q$ (I, § 6, Exercise 4) is isomorphic to the multiplicative group of the eight quaternions $\pm 1, \pm i, \pm j, \pm k$ in the quaternion algebra of type $(-1, -1)$ over a field in which $2 \neq 0$. Show that the algebra of the group $Q$ over a field $K$ in which $2 \neq 0$ is isomorphic to the product of four fields isomorphic to $K$ and the quaternion algebra of type $(+1, -1)$ over $K$ (if $c$ is the element of $Q$ corresponding to the quaternion $-1$, the elements of $Q$ can be written as $e, i, j, k, c, ci, cj, ck$; consider the basis of $E$ consisting of the elements $\frac{1}{2}(e + c), \frac{1}{2}(e - c), \frac{1}{2}(e + c)i, \frac{1}{2}(e - c)i, \frac{1}{2}(e + c)j, \frac{1}{2}(e - c)j, \frac{1}{2}(e + c)k, \frac{1}{2}(e - c)k$).