The Frölicher–Nijenhuis bracket and the geometry of $$G_2$$-and Spin(7)-manifolds

We extend the characterization of the integrability of an almost complex structure J on differentiable manifolds via the vanishing of the Frolicher–Nijenhuis bracket \([J, J]^{FN}\) to an analogous characterization of torsion-free \(G_2\)-structures and torsion-free \(\text{Spin(7) }\)-structures. We also explain the Fernandez–Gray classification of \(G_2\)-structures and the Fernandez classification of \(\text{Spin(7) }\)-structures in terms of the Frolicher–Nijenhuis bracket.