On real forms of JB*-triples

We introduce real JB^*-triples as real forms of (complex) JB^*-triples and give an algebraic characterization of surjective linear isometries between them. As main result we show: A bijective (not necessarily continuous) linear mapping between two real JB^*-triples is an isometry if and only if it commutes with the cube mappinga→a ³={aaa}. This generalizes a result of Dang for complex JB^*-triples. We also associate to every tripotent (i.e. fixed point of the cube mapping) and hence in particular to every extreme point of the unit ball in a real JB^*-triple numerical invariants that are respected by surjective linear isometries.