  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$^*$-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 $Y$.
