Why is this mapping surjective?

Question

It is mentioned in wikipedia that every single orthogonal $3 \times 3$ rational matrix is of the form

$$\dfrac{1}{m^2+n^2+p^2+q^2}\begin{pmatrix} m^2+n^2-p^2-q^2 & 2np-2mq & 2mp+2nq \\ 2mq+2np & m^2-n^2+p^2-q^2 & 2pq-2mn \\ 2nq-2mp & 2mn+2pq & m^2-n^2-p^2+q^2\end{pmatrix}$$

For $m,n,p,q \in \mathbb Q$. This statement refers to J. Cremona, Letter to the Editor, Amer. Math. Monthly 94. However in that letter it's only proved for real matrices with $m,n,p,q \in \mathbb R$. But it is stated (and left as exercise) that if we consider any subfield $\mathbb F \subset \mathbb R$ then all orthogonal matrices with elements from this subfield will be of the same form, i.e. same mapping from $\mathbb F^4 \to SO(3,\mathbb F)$ is also surjective. How to prove it?

Answer 1

This is an extended comment. (The answer is contained in the comment of Will Jagy: indeed it is a very simple argument that every rational orthogonal matrix is obtained from a rational quaternion).

The paper of J. Cremona referred in the question gives an incorrect formula for the recovery of the quaternion. The correct formula in Cremona's notation is $$\cos(\theta/2)+(b\mathbf{i}+c\mathbf{j}+d\mathbf{k})\sin(\theta/2),$$ where $(b,c,d)$ is the unit vector in the direction of the axis of rotation, and $\theta$ is the angle of rotation. It is far from evident from this formula that all rational rotations matrices arise from rational quaternions. But the simple direct argument of Pall does not use this inversion formula.

The fact that all rotation matrices with rational entries can be parametrized by the formula in question was discovered by Euler, Problema algebraicum ad affectiones prorsus singulares memorabile, Novi Commentarii Avcad. Sci. Imp. Petropolitanae, XV, 75-106 (Opera omnia, ser. 1, vol. 6, 287-315).

He also did the similar parametrization of rational $4\times 4$ rotations matrices.

All this was 72 years before the discovery of quaternions!

Let me cite Euler in English translation:

"This solution deserves full attention, as I arrived at it not using a definite method but rather by making guesses... If somebody can find a direct way to this solution, then it will have to be admitted that he made an outstanding contribution to analysis."

According to Khrushchev (Orthogonal polynomials and continued fractions, Cambridge UP, 2008), the first proof is due to D. Grave (1937).

In the book of Vilenkin Special functions and the theory of group representations, one can find a generalization to arbitrary dimension.