Maybe because that has required but not sufficienttly detailed parameters?

Assuming “intercept” is in the direction of “normal”, then you have a nice plane equation:
n.x * x + n.y * y + n.z * z - intercept = 0
Choose any four x,y,z triples which solve this and you got a quad.
Maybe pick one x,y,z on the plane which is “diameter” away from the footpoint (normal * intercept) and rotate that 90 degree around normal to get the second, mirror those two at the footpoint and you have a nice square.
If you give an “up” vector, you could also build it aligned to something.