A Le Page--Kaplansky theorem characterizing commutative JB*-triples

We prove that a Le Page-type inequality is also valid for metrically characterizing those JB${}^{\ast }$-triples that are commutative. More precisely, we establish that the following statements are equivalent for any JB${}^{\ast }$-triple $E$: $(a)$ $E$ is commutative. $(b)$ There exists $\gamma >0$ satisfying $$\parallel \{a,b,\{x,y,z\}\}\parallel \le \gamma \text{\hspace{0.17em}⁣}\parallel \{x,y,\{a,b,z\}\}\parallel ,\text{ for all }a,b,x,y,z\in E\mathrm{.}$$