Let $e$ be a tripotent in a JB$^*$-triple $\mathcal{U}$. Let $x\in \mathcal{U}$ and let $x_j=P_j(e)x$ for $j=1,2$. Then

-   $P_2(e)\{x,x,e\}\geq 0$ in $\mathcal{U}_2(e)$.

-   $\|\{x_j,x_j,e\} \|\leq \|P_2(e)\{x,x,e\} \| $ for $j=1,2$.

-   $\| x_j \|^2\leq 4 \| \{x_j,x_j,e\}\|$ for $j=1,2$.

The minimax principle {#sec: minmax}
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The main result of this section (Theorem \[t minmax\]) is a generalization of the minimax principle to the setting of weakly compact JB$^*$-triples. Generalizations of Weyl’s inequality and of the Cauchy-Poincaré (interlacing) theorem will be derived from this principle.

The following results can be considered as orthogonalising Gram-Schmidt processes in JB$^*$-algebras and JB$^*$-triples.

\[p sum of n minimal is rank n\] Let $\mathcal{A}$ be a JB$^*$-algebra. For every natural $n$, each collection of $n$ minimal tripotents in $\mathcal{A}$ is contained in $\mathcal{A}_2(u)$ where $u$ is a tripotent of rank less than or equal to $n$.

We shall proceed by induction. The case $n=1$ is trivial. Assume that the results holds for a given natural $n\in \mathbb{N}$ and take $n+1$ minimal tripotents $u_1,\ldots,u_{n+1} \in \mathcal{A}$. By the induction hypothesis we may assume that there exists a tripotent $e\in \mathcal{A}$ with rank less than or equal to $n$ such that $\{u_1,\ldots,u_n\}$ is contained in $\mathcal{A}_2(e)$. By 