Imam Ali (AS) did not use his authority as the head of the state for his own benefit or the benefit of his family, that is why he did not take back Fadak during his government. It is narrated that he was asked: Why don't you take back Fadak? He replied: The oppressor and the victim have left this life and Allah has given the victim her reward and the oppressor his punishment, so, I do not want to take it back.
Imam Ali (AS) spent the income of Fadak on the poor and needy despite of the fact that Fadak was the property of the inheritors of Fatimah (AS). Imam Musa Al-Kadhim (AS) was asked same question about Imam Ali (AS) not taking back Fadak for him and his children from Fatimah? He replied: We are Ahlul Bayt, take the right of people and give it to them, and do not take our own right and leave it to the Justice of Allah (SWT).
Keep in mind that one of the main objections of Muslims on Uthman ibn Affan that he used to give state properties to his relatives. If Imam Ali (AS) took Fadak , which was under the state properties since 25 years, and gave it to his children from Fatimah, the Munafiqeen will say that he is doing like Uthman. After all, he will give the income of it to the poor and needy, whether it is under his children ownership or state's ownership. He did not take it back during his government and left his right and his children's rights to Allah (SWT).