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$^*$-triples that are commutative. More precisely, we establish that the following statements are equivalent for any JB$^*$-triple $E$:
$(a)$ $E$ is commutative.
$(b)$ There exists $\gamma>0$ satisfying $$\big\|\{a,b,\{x,y,z\}\}\big\|\leq \gamma \ \! \big\|\{x,y,\{a,b,z\}\}\big\|, \hbox{ for all } a,b,x,y,z\in E.$$