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