Proposition \[p generalised extreme ray property\] we have that whenever $\|P_0(e)u_{n+1}\|>0$, $v=\frac{P_0(e)u_{n+1}}{\|P_0(e)u_{n+1}\|}$ is a minimal tripotent in $\mathcal{A}$, $e\perp v$ and $u_{n+1}$ belongs to $\mathcal{U}_2(e+v)$. Thus $u=e+v$ satisfy our desired statement.

Suppose now that $\|P_0(e)u_{n+1}\|=0$. Again by Proposition \[p generalised extreme ray property\] we can assume that $w=\frac{P_1(e)u_{n+1}}{\|P_1(e)u_{n+1}\|}$ is a minimal tripotent in $\mathcal{A}$, otherwise $u_{n+1}=P_2(e)u_{n+1}$ and hence $u=e$ gives our thesis. It can be derived from [@Kau97 Proposition 5.8] that every finite rank tripotent in a JB$^*$-algebra $\mathcal{A}$ is majorized by a unitary in $\mathcal{A}^{**}$. Let $\tilde{e}$ be a unitary in $\mathcal{A}^{**}$ with $e\leq \tilde{e}$ and let us denote $g=\tilde{e}-e$. Clearly $w\in \mathcal{A}^{**}_1(g)$ and $\{w,w,g\}$ is a positive (non-zero) element in $\mathcal{A}^{**}_2(g)\cap \mathcal{A}^{**}_1(w)$ (see [@FriRus85 Lemma 1.5]) whose range tripotent $v\in \mathcal{A}^{**}_2(g)\cap \mathcal{A}^{**}_1(w)$ has rank one or two (see [@DanFri87 Corollary 2.2]). If $rank(v)=2$ we have that $\{v,w,v\}$ is zero by Peirce arithmetics and a minimal tripotent by [@DanFri87 Proposition 2.1] which gives a contradiction. Therefore $v$ is a minimal tripotent in $\mathcal{A}^{**}_2(g) \cap \mathcal{A}^{**}_1(w)$ and $w\in \mathcal{A}^{**}_1(v)$. Actually, $v$ belongs to $\mathcal{A}$ by the Kadison Transitivity Theorem for JB$^*$-triples given in [@BunFerMarPer Theorem 3.3]. Finally, $u=e+v$ is a tripotent in $\mathcal{A}$ with rank less than or equal to 