The contributions in this note begin with a new characterization of (positive) scalar multiples of minimal tripotents in a general \(\hbox {JB}^*\) -triple E, proving that a non-zero element \(a\in E\) is a positive scalar multiple of a minimal tripotent in E if, and only if, its inner quadratic annihilator (that is, the set \(\phantom {a}^{\perp _{q}}\{a\} = \{ b\in E: \{a,b,a\} =0\}\) ) is maximal among all inner quadratic annihilators of single elements in E. We subsequently apply this characterization to the study of surjective additive maps between atomic \(\hbox {JBW}^*\) -triples preserving truncations in both directions. Assume that \(E = \bigoplus _{k\in \Gamma _1}^{\ell _{\infty }} C_k\) and \(F = \bigoplus _{{j\in \Gamma _2}}^{\ell _{\infty }} \widetilde{C}_j\) , are two atomic JBW \(^*\) -triples, where \(\{C_k\}_{k\in \Gamma _1}\) and \(\{\widetilde{C}_j\}_{j\in \Gamma _2}\) are two families of Cartan factors. Let \(A: E\rightarrow F\) be a surjective additive mapping between atomic \(\hbox {JBW}^*\) -triples, where E contains no one-dimensional Cartan factors as direct summands. We show that A preserves truncations in both directions if, and only if, there exists a bijection \(\sigma : \Gamma _1\rightarrow \Gamma _2\) , a bounded family \((\gamma _k)_{k\in \Gamma _1}\subseteq \mathbb {R}^+\) , and a family \((\Phi _k)_{k\in \Gamma _1},\) where each \(\Phi _k\) is a (complex) linear or conjugate-linear (isometric) triple isomorphism from \(C_k\) onto \(\widetilde{C}_{\sigma (k)}\) satisfying \(\inf _{k} \{\gamma _k \} >0,\) and \(\begin{aligned} A(x) = \Big ( \gamma _{k} \Phi _k \left( \pi _k(x)\right) \Big )_{k\in \Gamma _1},\ \hbox { for all } x\in E, \end{aligned}\) where \(\pi _k\) denotes the canonical projection of E onto \(C_k.\)
