tom.stoer criticized me and demanded that I "find how to relate [my] equations to the standard Lorentz transformation for spinors." (https://www.physicsforums.com/threads/klein-gordon-equation-and-particles-with-spin.563974/#post-3693864)

I did not feel that was a high-priority task, but recently I found a relativistically covariant form of the fourth-order partial differential equation for just one function that is generally equivalent to the Dirac equation. This form is applicable for an arbitrary component of the Dirac spinor (with some caveats) and any representation of gamma-matrices (with the standard hermiticity properties). The derivation was published in https://arxiv.org/abs/1502.02351 and (peer-reviewed) in "Quantum Foundations, Probability and Information" (A. Khrennikov, B. Toni, eds., Springer, 2018, pp.1-11).

Let me note that the relativistically covariant form is somewhat unusual and was not easy to derive. As the result was discussed at this forum, maybe its covariant form will be of interest for some people.