What is the ruling on taking possession of my deceased father's house that he bequeathed to me and my mother, knowing that there are no witnesses or written will, and that my father was married to another woman?

The house mentioned is an inheritance that must be divided among all heirs according to their legal share. The father's promise to gift the house or bequeath it to you is not legally binding. A bequest to an heir is invalid unless the other heirs approve it, and a gift is not binding unless possession is transferred during the lifetime of the giver. If the giver dies before possession is transferred, the gift becomes void. You are not entitled to exclusive ownership of the said house; rather, it belongs to all heirs, just like the rest of the inheritance.