In the literature I have seen the following definition of an inertial frame:
A frame is called inertial if any material point interacting with no other bodies or fields moves with constant velocity in a straight line with respect to this frame.
It is claimed that if another frame moves uniformly with respect to an inertial one, then it is also inertial.
In Newtonian mechanics that can be easily proved using the Galileo transformations.
Is there a more direct general way to see that without computations so that it would work simultaneously both in Newtonian mechanics and in special relativity?
Of course the above conclusion can be obtained similarly using the Lorentz transformations. However, usually in courses in special relativity the Lorentz transformations are deduced the other way around. The argument is based on the following two assumptions (among others): (1) in any inertial frame the speed of light is the same; (2) if a frame moves uniformly with respect to an inertial frame then it is inertial.
The second assumption is the focus of my question.