Is a woman considered lawful for her first husband, Islamically, if he took her back during her waiting period (iddah) without her knowledge, and he concealed the document of her return from her, and then she wanted to marry another man?

If the husband establishes proof of reconciliation (raj'ah) during the waiting period (iddah), then his reconciliation is valid, and the wife remains under his marital bond. Her marriage to another man, if it occurred, would be annulled. However, if he claims reconciliation without proof, and the wife denies it, then her word is to be accepted. It is commendable for the husband to inform his wife of the reconciliation, but even if he does not, the reconciliation is valid because it is a continuation of the existing marriage.