The conformal-Killing equation on G2G2- and Spin7-structures

Let MM be a 7-manifold with a G2G2-structure defined by ϕ∈Ω+3(M). We prove that ϕϕ is conformal-Killing with respect to the associated metric gϕgϕ if and only if the G2G2-structure is nearly parallel. Similarly, let NN be an 8-manifold with a Spin7-structure defined by ψ∈Ω+4(N). We prove that ψψ is conformal-Killing with respect to the associated metric gψgψ if and only if the Spin7-structure is parallel.