Here are some general comments on ownership that may be helpful - it not should be read as professional guidance.
Common Land is land that is privately owned which other people have private rights over e.g. for grazing - they have rights in common with others (abbreviated somewhat confusingly to common rights). All sorts of individuals and organisations own common land including county councils, lords of the manors, farmers, environmental organisations, MoD, water companies and private individuals. Common Land can be bought and sold just like any other land but will always remain as common land and be subject to the registered common rights as detailed on the Commons Register and other obligations summarised at https://www.gov.uk/owning-common-land. Strictly speaking an owner cannot have common rights over their own land but in many cases owners still have common rights registered in their name over the land they own. If an owner is only selling the common but not the farms to which the common rights are attached then the sale would not include those common rights, they stay attached to the land detailed in column 5 of the commons register.
One area that can be a bit confusing with local authorities is where they have assumed management control under s9 of the Commons Registration Act 1965. Under this local authorities may take action to protect commons where no owner is given on the Commons Register or the Land Registry but do not become the owner and therefore could not sell any land managed under that provision.