Extension of isometries from the unit sphere of a rank-2 Cartan factor

We prove that every surjective isometry from the unit sphere of a rank-2 Cartan factor C onto the unit sphere of a real Banach space Y, admits an extension to a surjective real linear isometry from C onto Y. The conclusion also covers the case in which C is a spin factor. This result closes an open problem and, combined with the conclusion in a previous paper, allows us to establish that every JBW∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {JBW}^*$$\end{document}-triple M satisfies the Mazur–Ulam property, that is, every surjective isometry from its unit sphere onto the unit sphere of a arbitrary real Banach space Y admits an extension to a surjective real linear isometry from M onto Y.