What is the ruling regarding the prayers performed by one who mistakenly believed that the Friday ghusl suffices for wudu', and does the statement of Ibn Taymiyyah and Al-Uthaymeen – that whoever omitted wudu' out of ignorance does not need to repeat the prayer except for the prayer of that time – apply to him? And is there sin upon the one who gave him that fatwa?

The relied-upon fatwa is that the Friday ghusl (ritual bath) does not suffice for wudu' (ablution). Therefore, your brother must repeat the prayers he performed. This is the view of the majority of scholars and is more cautious. Sheikh Al-Uthaymeen, may Allah have mercy on him, was of the opinion that repeating the prayer is more cautious in this situation. So, if someone performed ghusl for Friday prayer and prayed without wudu' out of forgetfulness, he must repeat the Friday prayer as Dhuhr (noon prayer), because the Friday ghusl is not a ghusl to remove ritual impurity (hadath), and impurity is not removed by it. However, if he performed ghusl for janabah (major ritual impurity) and did not perform wudu', his prayer is valid, with the necessity of rinsing the mouth and nose.