in a JB$^*$-triple $\mathcal{U}$. The symbol $\mathcal{U}_x$ will denote the JB$^*$-subtriple generated by $x$, that is, the closed subspace generated by all odd powers of the form $x^{[1]} := x$, $x^{[3]} := \{x,x,x\}$, and $x^{[2n+1]} := \{x,x,x^{[2n-1]}\},$ $(n\in \mathbb{N})$. It is known that $\mathcal{U}_x$ is JB$^*$-triple isomorphic (and hence isometric) to a commutative C$^*$-algebra in which $x$ is a positive generator (cf. [@Kau83 Corollary 1.15]). We identify the triple spectrum of $x$ with the spectrum of (the $C^*$-algebra) $\mathcal{U}_x$.

An element $e$ in a JB$^*$-triple $\mathcal{U}$ is said to be a *tripotent* if $\{e,e,e\} =e$. For each tripotent $e$ in $\mathcal{U}$ the eigenvalues of the operator $L(e,e)$ are contained in the set $\{0,1/2,1\},$ and $\mathcal{U}$ can be decomposed in the form $$\mathcal{U}= \mathcal{U}_{2} (e) \oplus \mathcal{U}_{1} (e) \oplus \mathcal{U}_0 (e),$$ where for $i=0,1,2,$ $\mathcal{U}_i (e)$ is the $\frac{i}{2}$ eigenspace of $L(e,e)$. This decomposition is known as the *Peirce decomposition*\[eq Peirce decomposition\] associated with $e$. The so-called *Peirce arithmetic* (also called Peirce rules) affirms that $\{\mathcal{U}_{i}(e),\mathcal{U}_{j} (e),\mathcal{U}_{k} (e)\}\subseteq \mathcal{U}_{i-j+k} (e)$ if $i-j+k \in \{ 0,1,2\},$ and $\{\mathcal{U}_{i}(e),\mathcal{U}_{j} (e),\mathcal{U}_{k} (e)\}=\{0\}$ otherwise, and $$\{\mathcal{U}_{2} (e),\mathcal{U}_{0}(e),\mathcal{U}\} =\{\mathcal{U}_{0} (e),\mathcal{U}_{2}(e),\mathcal{U}\} =0.$$ The projection $P_{k_{}}(e)$ of $\mathcal{U}$ onto $\mathcal{U}_{k} (e)$ is called the Peirce $k$-projection. It is 