Is it permissible for a brother to prevent his brother from building an apartment for himself on his private property in their deceased father's house, despite the mother's and sister's approval, using the argument that the house will not be sold as long as the mother is alive, or that it will be sold upon her death, or that he will demand his share in it, especially since the brother wishes to build at his own expense to preserve family ties and not relinquish their father's house?

If your father passed away and left the house, it becomes your property according to your legal shares, and the remaining heirs must respond if one of the heirs requests division. You can divide the house using one of the known types of division (partition by usufruct, by mutual consent, by lottery) if the house is suitable for division, or sell it and divide the price.

It is permissible for the heirs to agree to postpone the division, provided that all of them are mature. In this case, they become partners in the house, and none of them may dispose of it except with the permission of the others.

Based on this, you are not allowed to build an additional floor except with the consent of the other heirs. You have the right to demand the division of the house, and this is not considered an act of disobedience to your mother if leaving the matter as it is would cause you harm. However, it is better to leave things as they are, out of dutifulness to your mother and to preserve family ties, if possible.