A comment on the paper "Deformed Boost Transformations that saturate at the Planck Scale" by N.B.Bruno,G.Amelino-Camelia, and J.Kowalski-Glikman

An alternative (simplified) derivation of the dispersion relation and the expressions for the momentum-energy 4-vector $p_i,p_0$ given initially in [1] is provided. It has turned out that in a rather "pedestrian" manner one can obtain in one stroke not only the above relations but also the correct dispersion relation in $\omega-k_i$ space, consistent with the value of a velocity of a massless particle. This is achieved by considering the standard Lorentz algebra for $\omega-k_i$-space. A non-uniqueness of the choice of the time-derivative in the presence of the finite length scale is discussed. It is shown that such non-uniqueness does not affect the dispersion relation in $\omega-k_i$-space. albeit results in different dispersion relations in $p-p_0$-space depending on the choice of the definition of the time derivative.