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In this paper we examine interactions between (σ, τ) -derivations via commutator and consider new n-ary structures based on twisted derivation operators. We show that the sums of linear spaces of (σ^k, τ^l) -derivations and also of some of their subspaces, consisting of twisted derivations with some commutation relations with σ and τ, form Lie algebras, and moreover with the semigroup or group graded commutator product, yielding graded Lie algebras when the sum of the subspaces is direct. Furthermore, we extend these constructions of such Lie subalgebras spanned by twisted derivations of algebras to twisted derivations of n-ary algebras. Finally, we consider n-ary products defined by generalized Jacobian determinants based on (σ,τ) -derivations, and construct n-Hom-Lie algebras associated to the generalized Jacobian determinants based on twisted derivations extending some results of Filippov to (σ,τ) -derivations. We also establish commutation relations conditions for twisting maps and twisted derivations such that the generalised Jacobian determinant products yield (σ,τ,n) -Hom-Lie algebras, a new type of n-ary Hom-algebras different from n-Hom-Lie algebras in that the positions of twisting maps σ and τ are not fixed to positions of variables in n-ary products terms of the sum of defining identity as they were in Hom-Nambu-Filippov identity of n-Hom-Lie algebras.