If you look back at my edit of your wheel in perspective, as far as I know, the major axis can be found if you look at the square in perspective, and take the absolute width of the drawn square (without considering perspective). The major axis shouls lie there, because it divides the left and right part of the foreshortened square evenly. The minor axis is perpendicular to the major axis and you're done.

The only thing is that you need to draw a square in perspective in the first place. As of yet, I do not know a mathematical way for that.

The opposite, however—creating a square in perspective from an ellips—is an option. Currently, I can only imagine the steps, but I think they are correct:

1. Draw an ellips (narrow horizontally, wide vertically) and draw a vertical line A that is larger than the height of the ellips, and lean it on either side (it should hit either the left or right part of the ellips).

2. Create the square’s horizontal lines by drawing them from both ends of line A to a vanishing point to the opposite side while hitting the top and bottom of the ellips.

3. Create the square’s other vertical line by drawing it next to the other part of the ellips.

4. You’ve drawn a perfect square in perspective. This is for one-perspective. I think that more steps are needed for two-point (or three-point) perspective.

I will see what I can do later today. I could create examples for each mentioned step and see if my theory is correct. I do know that a circle (which with context is a sphere, obviously) should always fit perfectly inside a 3D cube (drawn on a 2D plane). That logic made me think that an ellips should always fit inside a square edge.

EDIT: I think this tutorial will help you a lot!:

https://youtu.be/h0HrmywKzFk

And this is the geometry behind it that you’re looking for:

https://www.handprint.com/HP/WCL/perspect3.html