If a husband says to his divorced wife, who is in her third menstruation after a single divorce, "Consider that I took you back into my wedlock ten days ago," is this considered a valid return, knowing that they have been completely separated and he has not touched her?

The phrase mentioned implies the possibility of reconciliation (raj'ah) but is not explicit about it. Scholars have differed on whether reconciliation occurs through an ambiguous statement (kinayah) accompanied by intention (niyyah). Ibn Qudamah holds that reconciliation occurs through an explicit verbal declaration, and its forms are "I have reconciled with you," "I have taken you back," "I have restored you," and "I have retained you." As for ambiguous statements (kinayah) like "I married her" or "He married her," there are two views: the first is that reconciliation does not occur through them, and the second is that it does occur through them if reconciliation was intended by them. Based on this, if you did not intend reconciliation with your statement, then reconciliation did not occur. However, if you did intend it, then reconciliation has occurred according to the view that it occurs through an ambiguous statement accompanied by intention.