Covariant four dimensional differential calculus in $\kappa$-Minkowski

It is generally believed that it is not possible to have a four dimensional differential calculus in $\kappa$-Minkowski spacetime, with $\kappa$-Poincar\'e relativistic symmetries, covariant under ($\kappa$-deformed) Lorentz transformations. Thus, one usually introduces a fifth differential form, whose physical interpretation is still challenging, and defines a covariant five dimensional calculus. Nevertheless, the four dimensional calculus is at the basis of several works based on $\kappa$-Minkowski/$\kappa$-Poincar\'e framework that led to meaningful insights on its physical interpretation and phenomenological implications. We here revisit the argument against the covariance of the four dimensional calculus, and find that it depends crucially on an incomplete characterization of Lorentz transformations in this framework. In particular, we understand that this is due to a feature, still uncovered at the time, that turns out to be fundamental for the consistency of the relativistic framework: the noncommutativity of the Lorentz transformation parameters. Once this is taken into account, the four dimensional calculus is found to be fully Lorentz covariant. The result we obtain extends naturally to the whole $\kappa$-Poincar\'e algebra of transformations, showing the close relation between its relativistic nature and the properties of the differential calculus.