Is it permissible for a committed man to sit with a divorced woman – a sitting that was the cause of her guidance – under the pretext of offering advice and consultation regarding her problems with her ex-husband, while not lowering his gaze, claiming pure intention and lack of temptation? And what is the evidence for this?

You have done well in advising your brother. The temptation of women is a great danger. Allah has commanded that questions be asked from behind a veil, as a purification for the hearts. This man should have offered his advice to the woman through her mahrams, not by sitting with her. Speaking with a non-mahram (definition) woman unnecessarily is an open door to temptation, and it places oneself in a position of suspicion. A Muslim avoids situations of doubt, as seen in the story of the Prophet, peace and blessings be upon him, with Safiyyah, may Allah be pleased with her. This also highlights the desirability of guarding oneself against people's ill-will and seeking safety.