  We prove that every commutative JB$^*$-triple has numerical index one. We
also revisit the notion of commutativity in JB$^*$-triples to show that a
JBW$^*$-triple $M$ has numerical index one precisely when it is commutative,
while $e^{-1}\leq n(M) \leq 2^{-1}$ otherwise. Consequently, a JB$^*$-triple
$E$ is commutative if and only if $n(E^*) =1$ (equivalently, $n(E^{**}) =1$).
In the general setting we prove that the numerical index of each JB$^*$-triple
$E$ admitting a non-commutative element also satisfies $e^{-1}\leq n(M) \leq
2^{-1}$, and the same holds when the bidual of $E$ contains a Cartan factor of
rank $\geq 2$ in its atomic part.

Operator Algebras;Functional Analysis