You have a misconception. The ellipse is a way to build a perspective image of a rectangular box. The ellipse is the same perspective view of a circle on a face of that box. You should build the perspective image of a rectangular face of a box around the ellipse, you should not try to crunch an ellipse into an already drawn 4-gon. That's because you can draw any ellipse inside a 4-gon and it will be a perfect perspective image of another ellipse.
In case your ellipse should be a perspective image of a circle on a square the question is valid. See NOTE1 and read my story to learn how the "Chord method" works.
The useful story would tell how one could construct a consistent cube if the horizon line and the perspective image of a circle on one of the faces is given. The story is a kind of advanced eyeballing for practical drawing on paper, it's not exact math. It doesn't start from the measures of the cube and imaging parameters, it starts from the horizon line and one ellipse.
In the next image we have the blue horizon line and an ellipse which presents a circle on one face of a cube. For simplicity we assume that the ellipse is actually a circle on a vertical face. In addition we assume 2 point perspective is good enough, so vertical lines are imaged as vertical:
The theory says that the minor axis of the ellipse points to the vanishing point of those horizontal edges of the cube which are perpendicular with the face of the ellipse (see NOTE2. We find the vanishing point (V) by extending the minor axis of the ellipse. Its easy in Illustrator which places nodes at the ends of the ellipse axles:
As already said we assume the ellipse is on a vertical face of the cube and we use 2 point perspective. The next image has the great eyeballing part:
The green lines are nothing special. They are all vertical lines (as we assumed vertical lines can be drawn as vertical lines) One goes through the midpoint of the ellipse and the other two are tangent lines. The great trick is to find the other vanishing point. We draw from an arbitary point A of the ellipse curve a straight line towards the horizon line so that the half segments ABD and CBD look visually as big. The line meets the horizon line at the wanted vanishing point U.
In drawing guidance texts this is the "Chord Method". The chord is the line AC.
Now we can draw a part of the edges of the face squares from the vanishing points and crossings:
The black lines are eyeballed tangents from U and the orange lines are drawn from V to the crossings.
We need still 2 lines to get a cube. One possibility is to use again an ellipse. We can eyeball one with vertical minor axis. It's the image of the circle inside the horizontal top square:
An eyeballed tangent from U and a vertical line from the crossing completes the cube.
I guess eyeballing sounds bad. I used Illustrator which should have precise snapping and I didn't utilize it except in the crossings. But: Chord method is designed for persons who draw with pen and paper. Some of them can also have a ruler and an ellipse drawing tool.
NOTE1: Chord method can also be used reversely to decide the lengths of the of the axles of the right ellipse (which is the image of a circle) if the vanishing points are already given.
You can find a longer description of the chord method from here: https://www.idsa.org/sites/default/files/2002_Randy%20Bartlett.pdf
Beware the "from math taken" but exact convention to define the direction of the planar surfaces by the directions of the surface normals. There the floor of an usual house is vertical and the walls are horizontal.
NOTE2: There are later comments which say that this rule is not a general perspective imaging fact. I do not know enough of perspective imaging math to say how accurate the rule is. So, take it as a practical way to find one of the possible images of possible fitting cubes, that one which produces an even looking result.