Someone has claimed that he has constructed a quaternion representation of the one dimensional (along the x axis) Lorentz Boost.
His quaternion Lorentz Boost is $v'=hvh^*+ 1/2( [hhv]^*-[h^*h^*v^*]^*)$ where h is (sinh(x),cosh(x),0,0). He derived this odd transform by substituting the hyperbolic sine and cosine for the sine and cosine in the usual unit quaternion rotation $v'=hvh^*$ and then subtracting out unwanted factors. You can see the short "proof" here:
I have argued that, whereas the quaternion rotations form a group, his newly devised transform probably does not. Two transformations of the form $v'=hvh^*+ 1/2( [hhv]^*-[h^*h^*v^*]^*)$ almost certainly do not make another transformation of the form $v'=fvf^*+ 1/2( [ffv]^*-[f^*f^*v^*]^*)$ He has not attempted to prove that and he won't because he thinks it is unnecessary.
He has responded that, even if I'm correct, he still has created a Lorentz Boost despite the fact that he has not created a group. I've argued that that it is essential that two Lorentz Boosts make another Lorentz Boost. Without that group structure there is no Boost. Is this correct?
Can you have a Lorentz Boost along the x axis without having the group structure Boost+Boost=Boost? (for the case of Boosts along the x axis. I know that, for boosts in different directions, Boost+Boost=Rotation)