Ameerul Momineen (a.s.) did not take back Fadak for two reasons
The skeptics claim that Ali (a.s.) should have taken back Fadak if it was indeed the property of Hazrat Fatima Zahra (s.a.). By leaving Fadak as it is, Ali (a.s.) has admitted that the property did not belong to Hazrat Fatima Zahra (s.a.).
The hollowness of such claims are exposed in many reports on Fadak, two of which are documented here.
a. Reward and punishment for the oppressed and oppressor
Imam Sadiq (a.s.) informs us why Ameerul Momineen (a.s.) did not take back Fadak on assuming caliphate.
He narrates: Ali (a.s.) did not take it back because the oppressor and the oppressed (Hazrat Fatima Zahra s.a.) had already set forth towards Allah, the Almighty. Allah had rewarded the oppressed (for patience) and had punished the oppressor (for usurping). Then Ali (a.s.) did not wish to take back something for which Allah had already punished the oppressor and had rewarded the oppressed.
Elal al-Sharae p 155 under Chapter of ‘Why Ameerul Momineen (a.s.) left Fadak as it is on assuming caliphate’
Awalim al-Uloom v 11 p 766
b. Ali (a.s.) would have reversed the judgements of the incompetent rulers
The past rulers were incompetent and did not have the requisite knowledge to rule the Muslims. If Ali (a.s) had it his way, he would have reversed many of their judgements as he declares in a lengthy sermon:Most certainly, if I order:
-the Maqam-e-Ibrahim to be restored to the very place in which the Messenger of Allah (s.a.w.a.) had put
-and give Fadak back to the inheritors of Fatima (s.a.)…(then Imam (a.s.) goes on to list a whole lot of blunders and innovations of the past rulers).
the people will abandon me (and rebel against me).
Al-Kafi v 8 p 58-63 under Sermon of Ameerul Momineen (a.s.)
Clearly, by not taking back Fadak. Ali (a.s.) wished for the chain of divine reward and punishment (for the oppressed and oppressor) to continue unbroken. Also, he (a.s.) was wary of a rebellion in his forces that he inherited from the incompetent rulers.