The inhomogeneous term in the 3-acceleration transformation law vanishes precisely for the subgroup of inertial frame transformations, identifying a well-defined equivalence class.

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3-acceleration(Relativistic mechanics): The acceleration of an object as described in a 3-dimensional spatial framework, denoted ξ²_α; its transformation law is more complicated and inhomogeneous compared to 4-acceleration.
Equivalence class(the bow and stern are different parts, but they're in the same equivalence class because they're both parts of one ship): A group of things that are considered the same or equal in some way—in this case, all the parts of the ship that belong to the same object.
Inertial frame(as used in physics): A reference point or perspective from which you measure motion, where objects at rest stay at rest unless something pushes them (like measuring from the ground versus from a moving train).
Transformation law(mathematics and physics): A rule that describes how something changes when you switch to a different measurement system or coordinate system.

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