Harmonic maps into the exceptional symmetric space G₂/SO(4)

We show that a harmonic map from a Riemann surface into the exceptional symmetric space G₂/SO(4) has a J₂ -holomorphic twistor lift into one of the three flag manifolds of G₂ if and only if it is ‘nilconformal’, that is, has nilpotent derivative. Then we find relationships with almost complex maps from a surface into the 6-sphere; this enables us to construct examples of nilconformal harmonic maps into G₂/SO(4) that are not of finite uniton number, and that have lifts into any of the three twistor spaces. Harmonic maps of finite uniton number are all nilconformal; for such maps, we show that our lifts can be constructed explicitly from extended solutions.