Map from unit quaternions to SO(3)?

Question

On the wikipedia page for "Rotation Group SO(3)" I read that there is a 2:1 surjection from the unit quaternions, $q=w+xi+yj+zk$, to the rotatation matrix $$Q= \left( \begin{array}{ccc} 1-2y^2-2z^2 & 2xy-2zw & 2xz+2yw \\ 2xy+2zw & 1-2x^2-2z^2 & 2yz-2xw \\ 2xz-2yw & 2yz+2xw & 1-2x^2-2y^2 \end{array} \right) $$ What I want is the derivation/proof of this surjection, or a source providing it. Thank you!

Answer 1

an ordinary vector $(a,b,c)$ in $\mathbb R^3$ is made into a quaternion with $0$ real part as $ v = ai + bj + ck.$ Then, given any unit quaternion $q,$ we get a new quaternion with real part $0$ from $q v \bar{q}.$ This turns out to be a linear isometry and orientation preserving. The map to $SO_3$ takes both $q$ and $-q$ to the same rotation.

The earliest I have seen this (including your matrix) is in number theory, specifically Gordon Pall, On The Rational Automorphs of $x_1^2 + x_2^2 + x_3^2,$ Annals of Mathematics, volume 41, 1940. However, it seems unlikely that this is the earliest appearance of the construction with real coefficients. Looked it up, the Hurwitz quaternions in number theory appeared in his 1919 book. Here, the coefficients are either all integers or all integers plus a half.