We use the octonionic multiplication ⋅ of S7 to associate, to each unit normal section η of a submanifold M of S7, an octonionic Gauss map γη:M→S6, γη(x)=x−1⋅η(x), x∈M, where S6 is the unit sphere of T1S7, 1 is the neutral element of ⋅ in S7. Denoting by N(M) the vector bundle of normal sections of M and by E(M) the vector bundle of sections of the vector bundle of endomorphisms of TM eqquiped with the Hilbert–Schmidt metric and defining the bundle homomorphism B:N(M)→E(M) by B(η)=Sη, where Sη is the second fundamental form of M determined by η, we prove that if M is a minimal submanifold of S7 and η∈N(M) is unitary and parallel on the normal connection, then γη is harmonic if and only if η is an eigenvector of B⁎B:N(M)→N(M), where B⁎ is the adjoint of B. If M is an isoparametric compact minimal submanifold of codimension k of S7 then B⁎B has constant non negative eigenvalues 0≤σ1≤⋯≤σk and the associated eigenvectors η1,⋯,ηk form an orthonormal basis of N(M), parallel on the normal connection, such that each γηj is an eigenmap of M with eigenvalue 7−k+σj. Moreover, σj=‖Sηj‖2, 1≤j≤k.