$u$ is a minimal projection in $\mathcal{U}_2(e)\cap \mathcal{U}_1(v)$, $u$ and $v$ are collinear by [@DanFri87 Proposition 2.1] and $\{v,v,e\}$ is a multiple of $u$. Using this information and Peirce arithmetics we get that $\|\{v,v,e\}\|u=\{v,v,e\}=\{v,v,u\}+\{v,v,e-u\}=\frac 12 u+\{v,v,e-u\} $ and hence $\{v,v,e\}=\frac 12 u= \frac 12 \{u,u,e\}$.

b\) By our assumptions, $\mathcal{U}_2(w)$ is a JB$^*$-algebra of rank 2 with involution $Q(w)$ (a conjugate linear automorphism of order 2). Let $x$ be a norm-one element in $\mathcal{U}_2(e)\cap \mathcal{U}_2(w)$. Clearly $Q(w)x$ is also a norm-one element orthogonal to $x$ by Peirce arithmetics. This implies that the rank of both $x$ and $Q(w)x$ has to be one and hence $x$ is a minimal tripotent in $\mathcal{U}_2(w)$ and also in $\mathcal{U}$. We define $u=x$.

By [@DanFri87 Proposition 2.1] we have that $(u,w,Q(w)u)$ form a trangle and clearly $w=w_1+w_2$ is the sum of two orthogonal minimal tripotents in $\mathcal{U}_2(w)$, where $w_1=\frac 12(u+w+Q(w)u))$ and $w_2=\frac 12(-u+w-Q(w)u))$. Finally, having in mind the Peirce arithmetics, we have that $Q(w)u\in \mathcal{U}_0(e)$ and $\{w,w,e\} =P_2(e) \{w,w,e\}=P_2(e)\{w_1,w_1,e\}+P_2(e)\{w_2,w_2,e\}= \frac{1}{2} (\{u,u,e\}+\{w,w,e\})$ which shows that $\{u,u,e\}=\{w,w,e\}$.

Throughout this paper we will make repeatedly use of Proposition 2.4 and Lemma 2.6 in [@BunFerMarPer], results that we summarize in the following proposition.

\[p normas sub u en Lusin\]

