Can we find all subfields of the quaternions $H( \mathbb{R})$?

Question

Let $R$ be a commutative ring. Define the Hamilton quaternions $H(R)$ over $R$ that is,

$$H(R)=\{a_0+a_1i+a_2j+a_3k\;\;:\;\;a_l\in R\}.$$

and multiplication is defined by: $i^2=j^2=k^2=ijk=-1$.

Now can we find all subfields of $H( \mathbb{R})$?

Answer 1

Two elements $x,y\in H(\mathbb{R})$ commute iff their imaginary parts are linearly dependent over $\mathbb{R}$. So any subfield of $H(\mathbb{R})$ is contained in one of the subfields of the form $\mathbb{R}+\mathbb{R}v$ where $v$ is purely imaginary; every such subfield is isomorphic to $\mathbb{C}$. Classifying subfields of $\mathbb{C}$ is not particularly feasible, so this is as much as you can hope to say.