In order to gain the respect of any of your lieutenants, you first have to give them enough districts of the city, at least three. After that, you must complete part of their side missions, which will improve your relations with the lieutenants. This will give you the opportunity to learn more about their history and personal life. All three stories are tragic in their own way, and they tell a lot about the motivation of the characters. It is worth adding here that loyalty status will not bring you any practical benefit - you have not received any bonuses or improvements. It's possible that a good relationship with the lieutenants influences the ending.
