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

Fr\"olicher-Nijenhuis bracket and geometry of $G_2$-and ${\rm Spin}(7)$-manifolds