In The Quaternion Group and Modern Physics by P.R. Girard, the quaternion form of the general relativistic equation of motion is derived as:

$du'/ds = (d a / d s ) u {a_c}^* + a u ( d {a_c}^* / d s ) + a (du/ds ) {a_c}^*$

where $u$ is a "minquat" of the form $(ct,ix,iy,iz)$ and a is an arbitrary function satisfying $a{a_c}=1$

This is said to "correspond" to the equation of motion of GR. It is an incredibly elegant formulation of the equation.

I've tried applying this formula to $a=\cos(s)-i\sin(s)$ and, unless I've made some errors-and I probably did, this results in $x(1+\cos(2s))+ ict(1+\sin(2s))$ for the first two terms.

1.What is the physical significance of these terms?

2.Does this really represent the GR equations of motion?

3.What other functions satisfying the condition $a{a_c}=1$ should I use? (I've also used $e^{i\theta}$ and this switched $x$ and $ct$ which is interesting, but-again-what does it actually represent?)

4.Can I use this equation to derive any standard results? I would like to see the centrifugal and other fictitious forces come out of it maybe.

I've found that the Runge Lenz vector is used to calculate orbits in Newtonian Gravity and GR, but I'm baffled by the introduction of this vector in this case.

Can it be translated into a standard formulation of the equations of motion?

Any information or advice for tackling this would be appreciated.