Fr\"olicher-Nijenhuis bracket and geometry of $G_2$-and ${\rm   Spin}(7)$-manifolds
  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. 
Kotaro Kawai, H\^ong V\^an L\^e and Lorenz Schwachh\"ofer
math.DG math-ph math.MP