Sam: after playing with quaternion multiplication for about 10-15 minutes, I am on the fence about the conjecture relating to the product of two quaternions.

The notation I am using has two sets of unknowns: (A, B, C, D) and (a, b, c, d).

There are 4 equations in the 8 unknowns.
  • Aa - Bb - Cc - Dd = 0
  • Ab + Ba + Cd - Dc = 0
  • Ac + Ca + Db - Bd = 0
  • Ad + Da + Bc - cB = 0
Find a solution to the above equations such that neither (A, B, C, D) nor (a, b, c, d) is (0, 0, 0, 0) and you have two non zero quaternions whose product is zero.

4 equations in 8 unknowns would seem to have many solutions, even with the restrictions given. There seems to be a lot of degrees of freedom here to play with. This makes me think there are solutions meeting the conditions.

After working for about 10-15 minutes, it seems as though there are symmetries to the equations which start forcing unknowns to be zero. This makes me think there are no solutions meeting the conditions.

Perhaps you or somebody else would like to spend some time on this. My lazy tendencies have won out over my obsessive-compulsive personality.