Given a non-inertial frame of reference what is(are) the condition(s) required to affirm whether another frame of reference (being observed from the current non-inertial frame) is inertial or non-inertial?
I think a brief background to the question is required. I thought of this situation while considering the following case: suppose we are observing an observer (in space) from Earth, how may I claim that the the reference frame attached to that observer is inertial or not? Clearly earth is a non-inertial frame of reference, hence the question.
I assume you can't go to the space observer's frame of reference, so you have to do it at a distance.
Motion is more generally composed of inertial motion (field forces) and non-inertial motion (contact forces). In your example, you are on the surface of Earth observing someone in space, so you have to separate the components.
When you are in deep-space (gravity negligible) either "standing still" or travelling with constant velocity, then you are clearly in an inertial frame. The same hapens when you are in free fall in a gravitational field (either in a parabolic trajectory over a planet's surface, or in orbit around a large mass). In both scenarios, you don't feel motion, you are weightless, because gravity is a field force and thus your accelerometers measure zero, and you are always oriented to the same direction in space, just as a gyroscope.
The inertial motion due to gravity must be measured by an external referential, say the distant stars.
Then, you have to know how much of your motion is non-inertial. Since you are being held in the surface of Earth by the normal force, then you can feel weight, because the normal force is a contact force, and thus you can measure the proper acceleration it causes using an accelerometer.
When you have determined how is your motion composed, measuring both your proper acceleration due to the normal force using accelerometers, and the coordinate acceleration due to gravity using relative position to distant stars, then you have to measure the motion of the observer in space relative to you.
Now the trick part: the observer in space motion's may also be composed by inertial and non-inertial movement, so you have to estimate what the gravity field looks like in the observer in space's position.
Once you have a) your motion decomposed into inertial and non-inertial components, b) the relative motion between you and the observer in space mapped in detail - so you know what their movement relative to the distant stars is, and c) an estimate of the gravity field around the space observer, that is, what are the geodesics in their surroundings, then you can d) subtract their inertial motion, and whatever motion is left is their non-inertial motion.
Note that this is very difficult to do in practice, not only beacuse the form of the gravitational field may be very complicated (many celestial objects, near and far, dust clouds, small but heavy meteorites, etc.), but because so many factors can cause the space observer's proper motion (small amounts of gases being expelled from it, thermal radiation emission and absorption, and even anisotropic radiation pressure - see the Pioneer anomaly). See, for example, the controversy surrounding the non-gravitational trajectory of the ʻOumuamua object.
