  An experiment in which the Clauser-Horne-Shimony-Holt inequality is maximally
violated is self-testing (i.e., it certifies in a device-independent way both
the state and the measurements). We prove that an experiment maximally
violating Gisin's elegant Bell inequality is not similarly self-testing. The
reason can be traced back to the problem of distinguishing an operator from its
complex conjugate. We provide a complete and explicit characterization of all
scenarios in which the elegant Bell inequality is maximally violated. This
enables us to see exactly how the problem plays out.
