Exercise 5, § 1

a) Let $K$ be a non-commutative division subring of the division ring of quaternions $\mathbf{H}$; show that the centre $Z$ of $K$ is a subfield of the centre $\mathbf{R}$ of $\mathbf{H}$. (Observe that every element of $K$ commutes with every element of the field $L$ generated by $Z \cup \mathbf{R}$; if $Z \not\subset \mathbf{R}$, the field $L$ would be a maximal subfield of $\mathbf{H}$, and $K$ would be contained in $L$, contrary to hypothesis.)

b) Deduce that every isomorphism $f$ (not necessarily continuous) of $\mathbf{H}$ onto a division subring of $\mathbf{H}$ is an inner automorphism $x \to axa^{-1}$ of $\mathbf{H}$. [The restriction of $f$ to $\mathbf{R}$ is an isomorphism of $\mathbf{R}$ onto a subfield of $\mathbf{R}$, by virtue of $a$); use Exercise 3 of Chapter IV, § 3.]