that $\varphi (z) =1$, it follows from [@BarFri Proposition 1.2] that the assignment $$(x,y)\mapsto \varphi\{x,y,z\}$$ defines a positive sesquilinear form on $W,$ the values of which are independent of choice of $z$, and induces a prehilbert seminorm on $W$ given by $$\|x\|_{\varphi}:= \left(\varphi\ \{x,x,z\}\right)^{\frac{1}{2}}.$$ As $\varphi$ ranges over the unit sphere of $W_*$ the topology induced by these seminorms is termed the strong\*-topology of $W$. Moreover, $\| \cdot \|_{\varphi}$ is additive on the sum of orthogonal elements (see [@FerPer06 Lemma 3.3] for a more general result).

Whenever $\mathcal{U}$ is a weakly compact JB$^*$-triple, associated to every minimal tripotent $v$ in $\mathcal{U}$ there exists, $\varphi_v$, an (unique) extreme point in the closed unit ball of $\mathcal{U}^*$ satisfying $\varphi_v=\varphi_v\circ P_2(v)$ and $\varphi_v(v)=1$ (cf. [@FriRus85 Proposition 4]). Therefore we can define the seminorm $\| \cdot\|_v=\|\cdot\|_{\varphi_v}$ in $\mathcal{U}$. Since $v$ is minimal and $P_2(v)\{x,x,v\}$ is positive in $\mathcal{U}_(v)=\mathbb{C}v$, we also have that $$\| x\|_v^2=\|x\|_{\varphi_v}^2= \varphi_v\circ P_2(v)\{x,x,v\}=\|P_2(v)\{x,x,v\}\|.$$ This particular seminorms will be widely used throughout the present work and where introduced with more generality in [@BunFerMarPer].

Let $u,v$ be tripotents in a JB$^*$-triple $\mathcal{U}$. We say that $u$ and $v$ are *collinear* (written $u\top v$) if $u\in \mathcal{U}_1(v)$ and $v\in \mathcal{U}_1(u)$. We say that $u$ 