Physics books rarely make the distinction between active or passive Lorentz transformations. The usual Lorentz transformations of the spacetime coordinates in two different inertial frames seem to me to be passive transformations, because by definition passive transformations are coordinates transformations; but, we also say that spacetime coordinates transform as a 4-vector (a 4-vector is by definition a collection of entries that transform just as the coordinates do) and active transformations are those transformations that act on the vectors. Grouping the four spacetime coordinates in a 4-vector seems to blur the distinction between active and passive Lorentz transformations. Also, what do we mean precisely when we say that energy and momentum have the same transformation property of the coordinates (i.e. form a 4-vector)? A Lorentz transformations act on Minkowki spacetime M, whose elements are spacetime events, right? So, where do general 4-vector live? And "how" do Lorentz transformations act on them? To sum it all up, the physically obvious concept of "change of reference frame" would be called active or passive by a mathematician?