Maps preserving the truncation of triple products on Cartan factors

Let {C_i}_i∈Γ_1, and {D_j}_j∈Γ_2, be two families of Cartan factors such that all of them have dimension at least 2, and consider the atomic JBW^*-triples A=⊕_i∈Γ_1^ℓ_∞ C_i and B=⊕_j∈Γ_2^ℓ_∞ D_j. Let Δ :A → B be a (non-necessarily linear nor continuous) bijection preserving the truncation of triple products in both directions, that is, a {b,c,b}⇔Δ(a) {Δ(b),Δ(c),Δ(b)} Assume additionally that the restriction of Δ to each rank-one Cartan factor in A, if any, is a continuous mapping. Then we show that Δ is an isometric real linear triple isomorphism. We also study some general properties of bijections preserving the truncation of triple products in both directions between general JB^*-triples.