So I actually was working on these 2 years ago!
This link discusses the quaternions and the search for a triplex combination
http://www.songho.ca/math/quaternion/quaternion.htmlThe author sets out to demonstrate by contradiction that a triple form does not exist for the premise he starts with.
2 things: contradiction is not a proof of none existence! And the base elements chosen are not the only possible basis.
When I tackled this issue it was from a different point of view. I was looking at group behaviours modulo (n), and specifically applying that to the rotation of the axes of a reference frame. I could see what I was doing, but I could not explain what I was doing!
I remember being very poorly at the time, recovering from a life threatening blood disorder. Working on this issue was a welcome distraction.
Applying the roots of unity in this way as a control for the axes interchange seemed an interesting thing to do, but I honestly had no clue what I was doing and moved from one inspirational flash to another.
Today I have the benefit of the mentorship of Hermann Grassmann and that has made a huge difference in terms of my ability to coherently understand what I was doing back then. It is making a huge difference going forward also.