satisfying $w \vdash v_{11}$, $w \vdash v_{22}$ and $v_{11}\perp v_{22}$ such that $$v = \alpha v_{11}+\beta  w +\delta v_{22}.$$

Notice that in both cases described above, $|\alpha|=\|P_2(e)v\|$, $|\delta| =\|P_0(e)v\|$ and $(|\alpha|+|\delta|)^2\leq 1$.

\[l extreme ray prop en Peirce 1\] Let $\mathcal{U}$ be a JB$^*$-triple and let $e$ be a tripotent in $\mathcal{U}$. Then

-   For every minimal tripotent $v$ in $\mathcal{U}_1(e)$ we have that $u=2\{v,v,e\}$ is a minimal projection in $\mathcal{U}_2(e)$ and $2\{v,v,e\}=\{u,u,e\}$.

-   For every rank-2 tripotent $w$ in $\mathcal{U}_1(e)$ with $\mathcal{U}_2(e)\cap \mathcal{U}_2(w)\neq \{0\} $ there exists a minimal tripotent $u \in \mathcal{U}_2(e)$ such that $\{w,w,e\}=\{u,u,e\}$.

a\) Given a minimal tripotent $v$ in $\mathcal{U}_1(e)$ we have that $v,e$ are compatible, i.e. their Peirce projections commute (see [@FriRus85 Lemma 1.10]), thus Peirce arithmetics give $P_2(v)e=0$ and $\{v,v,e\}$ is an element in $\mathcal{U}_2(e)\cap \mathcal{U}_1(v)$ positive in $\mathcal{U}_2(e)$ and not zero by [@FriRus85 Lemma 1.5]. We denote by $u$ the range projection of $\{v,v,e\}$ in $\mathcal{U}_2^{**}(e)$ so $u\in \mathcal{U}_1^{**}(v)$. We recall that $\mathcal{U}^{**}_1(v)$ is a JBW$^*$-triple of rank smaller than or equal to 2 [@DanFri87 Corollary 2.2]. If we assume that $u$ has rank two, Peirce arithmetics show that $\{u,v,u\}=0$ while [@DanFri87 Proposition 2.1] gives $\|\{u,v,u\}\|=1$ which leads to a contradiction. Therefore 