find results in case of Euclidean Jordan algebras, which happen to be finite dimensional JB-algebras by [@FarKor Page 42] and [@HanStor 3.1.7]. The first stunning result in this area is the minimax principle obtained by U. Hirzebruch in 1970 [@Hir]. The implications of this result seemed to be unnoticed for almost forty years until M.S. Gowda, J. Tao and M. Moldovan used it to derive a Weyl’s perturbation inequality in Euclidean Jordan algebras (see [@GowTaoMol Theorem 9]). Gowda and Tao also obtained a Cauchy-Poincaré interlacing theorem in [@GowTao]. Unfortunately, unlike in the case finite dimensional C$^*$-algebras, these results can not be used directly to derive their analogs for non-necessarily symmetric elements in a finite dimensional JB$^*$-algebra.

In Section \[sec: minmax\] we obtain a generalized minimax principle for weakly compact JB$^*$-triples (see Theorem \[t minmax\]). In this geometric characterization of the singular values of an element, arranged in decreasing order and counting multiplicity, the role played by the subspaces and the elements of the Hilbert space is now played by tripotents and minimal tripotents respectively. As consequences of this result a Weyl inequality and a Cauchy-Poincaré interlacing Theorem are also obtained.

Section \[sec: Ky Fan\] is entirely devoted to present a 