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I forgot to mention:

if the elevator (rocket) accelerates under the influence of an gravitational field (at the turnaround) then what would we have ????

an inertial frame that accelerates ??

Yes. There are two different notions of "acceleration": coordinate acceleration, and proper acceleration. Coordinate acceleration depends on your coordinate system. For example, the path: [itex](x=0, y=vt)[/itex] has zero coordinate acceleration in Cartesian coordinates. But if you switch to polar coordinates: [itex]r = \sqrt{x^2 + y^2}[/itex], [itex]\theta = tan^{-1}(\frac{y}{x})[/itex], then the coordinate acceleration in terms of [itex]r[/itex] and [itex]\theta[/itex] is nonzero. Proper acceleration is what is measured by an accelerometer, and it is independent of what coordinate system you use.

even from the point of view of special relativity, this has to be a IRF since the laws of stay unchanged in their simplest form.