Harmonic maps into the exceptional symmetric space $G_2\mathrm{/}SO(4)$

We show that a harmonic map from a Riemann surface into the exceptional symmetric space $G_2\mathrm{/}\mathrm{S}O(4)$ has a $J_2$-holomorphic twistor lift into one of the three flag manifolds of $G_2$ if and only if it is `nilconformal’, i.e., 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 $G_2\mathrm{/}\mathrm{S}O(4)$ which are not of finite uniton number, and which 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.