Cartan factor* is formed by the $t$-symmetric operators. A Banach space $X$ is called a *Cartan factor of type 4* or *spin* if $X$ admits a complete inner product $(.|.)$ and a conjugation $x\mapsto \overline{x},$ for which the norm of $X$ is given by $$\|x\|^2 = (x|x) + \sqrt{(x|x)^2 -|
(x|\overline{x}) |^2}.$$ *Cartan factors of types 5 and 6* (also called *exceptional* Cartan factors) are both finite dimensional and coincide with the $1\times 2$-matrices and symmetric $3\times 3$-matrices over the complex Cayley numbers, respectively.

We recall now some terminology taken from [@BunChu92]. Let $K(H,H')$ be the space of all compact linear operators between two complex Hilbert spaces. We shall write $K(H)$ instead of $K(H,H)$. If $C_j$ is a Cartan factor of type $j\in \{1,2,3,4,5,6\}$, we define $K_1 = K(H,H')$ for $C_1= L(H,H')$, $K_j=C_j\cap K(H)$ for $j=2,3$, and in the remaining cases $K_4 = C_4,$ $K_5 = C_5$, and $K_6 = C_6$. The JB$^*$-triples $K_1,K_2,\ldots, K_6$ are called elementary JB$^*$-triples. The class of *weakly compact JB$^*$-triples* consist of those JB$^*$-triples which are a (possibly infinite) $c_0$-sum of elementary JB$^*$-triples (see [@BunChu92 Theorem 2.4]).

Given a JBW\*-triple $W,$ a norm-one element $\varphi$ of $W_*$ and a norm-one element $z$ in $W$ such 