Almost complex structures in 6D with nondegenerate Nijenhuis tensors and large symmetry groups

For an almost complex structure $J$ in dimension 6 with nondegenerate Nijenhuis tensor $N_J$, the automorphism group $G=Aut(J)$ of maximal dimension is the exceptional Lie group $G_2$. In this paper we establish that the sub-maximal dimension of automorphism groups of almost complex structures with nondegenerate $N_J$, i.e. the largest realizable dimension that is less than 14, is $\dim G=10$. Next we prove that only 3 spaces realize this, and all of them are strictly nearly (pseudo-) K\"ahler and globally homogeneous. Moreover, we show that all examples with $\dim Aut(J)=9$ have semi-simple isotropy.