Is a business registered in the name of the eldest son, and the profits generated from it, considered a gift from the father to him and his brother working with him, or is it compensation for their efforts that does not fall under the inheritance, especially since the father bought lands from these profits and registered them in his own name to be distributed among the heirs?

The principle is that all assets registered in your name, such as businesses and cars, are included in the inheritance unless a gift (hiba) is proven. This is not considered compensation for your work and effort, as you were benefiting and living from it. If your work was a voluntary contribution, the matter is clear. If it was for compensation, then you are entitled to a fair wage, from which what you have already taken will be deducted. The remainder will be returned to you before the inheritance is divided, if you intend to claim it. The father must be fair in giving gifts among his children. If he is not fair, then all gifts must be returned to the inheritance and divided according to Islamic law, unless the preference was made with the consent of all adult and mature heirs.