The most important thing to remember when trying to help ease eczema is not as flexible as possible. You may well need to vary the oils and change the way you use them, since the nature of the condition can be due to so many factors, and the skin can get better for a while with one oil and then may need something different as the symptoms change.
Some of the most useful oils to use on the skin are Chamomile, Geranium, Lavender, Lemon Balm, and Rose. They should be used in at most a 1 percent dilution initially and can be incorporated in a light aqueous cream, or perhaps use a thicker cream, ointment or pure vegetable oil base if the skin is extremely dry or weakened. Try these oils one at a time at first, to see whether the skin reacts to any of them.