our original minimal tripotent.

We proceed now with the general case. As we have just seen, for every minimal tripotent $u_i$ we have associated another tripotent $v_i$ in $\mathcal{U}_2(e)$ minimal or zero. We give raise to the tripotent $u$ by using Proposition \[p sum of n minimal is rank n\], having in mind that $v_1,\ldots,v_n$ are minimal tripotents (or zero) in the JB$^*$-algebra $\mathcal{U}_2(e)$. The desired statement can be checked using Peirce arithmetics.

Note that the result above is trivially satisfied when $e$ is a finite rank tripotent of $rank(e)\leq n$ by taking $u=e$. The following corollary is an orthogonalized version of Lemma \[l pre-minimax\].

\[c gram schmidt\] Let $\mathcal{U}$ be a JB$^*$-triple. Let $f$ be a tripotent in $\mathcal{U}$ with $rank(f)=n$. Then for every tripotent $e$ in $\mathcal{U}$ with $rank(e)=m> n$ there exists $u$ in $\mathcal{U}_2(e)$ a tripotent with $rank(u)= m-n$ such that $u\perp f$.

Every element in a weakly compact JB$^*$-triple can be decomposed as a (possibly infinite) linear combination of mutually orthogonal minimal tripotents. Concretely, given a weakly compact JB$^*$-triple, $\mathcal{U}$, for every $x \in \mathcal{U}$ there exist a sequence of mutually orthogonal minimal tripotents $(u_i)_{i\in \mathbb{N}}$ and a decreasing sequence of nonnegative real numbers $(\lambda_i(x))_{i\in \mathbb{N}}$ 