Is expiation (Kaffarah) required for intentional killing that is permissible by law?

Primary text

Expiation is not incumbent upon intentional killings that are permissible, such as the killing of a combatant (Harbi), a rebel (Baghi), an adulterer proven guilty (Zani Muhsan), or killing carried out as retribution (Qisas) or legal penalty (Hadd). This is because these acts are commanded, and expiation does not apply to acts that are commanded. This understanding is derived from the statement of Al-Khiraqi.