Under the fitting headline "Example of Garbage Mathematics" Evans repeats his claim of the Lorentz transformation of the unit vector in boost direction
k = k'
on his blog site. Since Evans likes to refer to S.M. Carroll's Lecture Notes on General Relativity I recommend to Evans to have a look at the figure taken from p.7 where for a Lorentz boost in x/x'-direction the corresponding unit vectors i and i' are displayed: Evidently both vectors are different, so also k ≠ k' in case of a Lorentz boost in z/z'-direction.
for his blog note is fully justified.
No doubt, Evans has not understood the equations of Lorentz transform that are outlined by Carroll on p.43, eq. (2.11) together with eqs. (1.21) on p.6..
(02.10.2007) Further Correction of Bruhn in Evans' Collection of "Rebuttals"
(30.09.2007) Comments on Evans' Note 1 on the Lorentz transform
(25.05.2007) Commentary on Evans' Key Derivation 6
(09.05.2007) Remark on Evans' rebuttal (B(3) Theorem)