that the rank of the zero tripotent is 0.

A JBW$^*$-triple is a JB$^*$-triple which is also a dual Banach space (with a unique isometric predual [@BarTim86]). It is known that the second dual of a JB$^*$-triple is a JBW$^*$-triple (compare [@Din86]). An extension of Sakai’s theorem assures that the triple product of every JBW$^*$-triple is separately weak$^*$-continuous (cf. [@BarTim86] or [@Horn87]).

Another illustrative examples of JBW$^*$-triples are given by the so-called Cartan factors, whose classification generalize that for finite dimensional JB$^*$-algebra factors given by P. Jordan, J. von Neumann and E. Wigner in [@JorNeuWig34]. We will present them in order to make the notion of weakly compact JB$^*$-triple more approachable.

A complex Banach space is a *Cartan factor of type 1* is it coincides with the complex Banach space $L(H, K)$, of all bounded linear operators between two complex Hilbert spaces, $H$ and $K$, whose triple product is given by .

Given a conjugation, $j$, on a complex Hilbert space, $H$, we can define a linear involution on $L(H)$ defined by $x \mapsto x^{t}:=j x^* j$. A *type 2 Cartan factor* is a subtriple of $L(H)$ formed by the skew-symmetric operators for the involution $t$; similarly, a *type 3 