The positive discriminant case of Nagell's theorem for certain cubic orders

It is proved that a real cubic unit u, whose other two conjugates are also real, is almost always a fundamental unit of the order Z[u]. The exceptions are shown to consist of a single infinite family together with one sporadic case. This is an analogue of Nagell's theorem for the negative discriminant case i.e. the case where u does not have any real conjugate.