known that Peirce projections are contractive (cf. [@FriRus85]) and satisfy that $P_{2}(e) = Q(e)^2,$ $P_{1}(e) =2(L(e,e)-Q(e)^2),$ and $P_{0}(e) =Id_E - 2 L(e,e) + Q(e)^2,$ where $Q(e):\mathcal{U}\to \mathcal{U}$ is the conjugate linear map given by $Q(e) (x) =\{e,x,e\}$. The Peirce subspace $\mathcal{U}_2(e)$ is a JB$^*$-algebra with Jordan product $x\circ y=\{x,e,y\}$ and involution $Q(e)$.

There is a partial order in the set of tripotents given by $e\leq f$ if and only if $P_2(e)f=e$ (equivalently, $f=e+P_0(e)f$ or $e$ is a projection in the JB$^*$-algebra $\mathcal{U}_2(f)$).

Two elements $x,y$ in $\mathcal{U}$ are orthogonal ($x\perp y$) whenever $\{x,x,y\}=0$ (see [@BurFerGarMarPer] for some equivalent definitions). When particularized to tripotents $e,f$ we have that $e\perp f $ if and only if $P_0(e)f=f$. A tripotent $e$ in $\mathcal{U}$ is said to be *minimal* if $\mathcal{U}_2(e)=\mathbb{C} e \neq \{0\}$ and *finite rank* if $e$ is the finite sum of mutually orthogonal minimal tripotents in $\mathcal{U}$. Analogously, an element in $\mathcal{U}$ is said to be of finite rank if it is a finite linear combination of mutually orthogonal minimal tripotents in $\mathcal{U}$. The *rank* of a finite rank element $x$, $rank(x)$, is the minimum number of mutually orthogonal minimal tripotents which can be used to express $x$. We shall consider 