*governs* $v$, $u \vdash v$, whenever $v\in \mathcal{U}_{2} (u)$ and $u\in \mathcal{U}_{1} (v)$.

We will refer to the following statement as the *extreme ray property* for JB$^*$-triples and combines some well-known results by Y. Friedman and B. Russo (see [@FriRus85 Proposition 5 and Proposition 6] and [@FriRus93 Proposition 1.5]) with the generation of Cartan factors by orthonormal grids presented in [@DanFri87].

\[p generalised extreme ray property\] Let $\mathcal{U}$ be a JB$^*$-triple. Given a minimal tripotent $v$ in $\mathcal{U}$, then for every arbitrary tripotent $e$ in $\mathcal{U}$, $P_2(e)v$ and $P_0(e)v$ are a multiple of some minimal tripotent in $\mathcal{U}$. More concretely, one of the following happens

1.  There exists $ \alpha, \beta, \gamma, \delta \in \mathbb{C}$ with $|\alpha|^2+|\beta|^2 + |\gamma|^2+ |\delta|^2=1$, $\alpha \delta  = \beta \gamma$ and minimal tripotents $v_{11},v_{12},v_{21},v_{22}$ (zero if the corresponding coefficient vanishes) satisfying $v_{11}\perp v_{22}$, $v_{12}\perp v_{21}$, $v_{11}\top v_{12}\top v_{22}\top v_{21}$, $v_{11}\in \mathcal{U}_2(e)$, $v_{21},v_{12} \in \mathcal{U}_1(e)$ and $v_{22}\in \mathcal{U}_0(e)$, such that $$v = \alpha v_{11}+\beta v_{12}+\gamma v_{21}+\delta v_{22}$$

2.  There exists $ \alpha, \beta, \delta \in \mathbb{C}$ with $|\alpha|^2+ 2|\beta|^2 + |\delta|^2=1$, $\alpha \delta  = \beta^2$, minimal tripotents $v_{11}\in \mathcal{U}_2(e)$, $v_{22}\in \mathcal{U}_0(e)$ and a tripotent $w\in \mathcal{U}_1(e)$ (each of these tripotents zero if the corresponding coefficient vanishes), 