Trivial minimal ideals of Jordan systems

We show that in Jordan systems (algebras, triple systems, and pairs) monomials containing two elements of a trivial minimal ideal vanish, so improving the answer given by Anquela and Cortés in [J.A. Anquela, T. Cortés, Minimal ideals of Jordan systems, Invent. Math. 168 (2007) 83–90] to the problem of Nam and McCrimmon in [N.S. Nam, K. McCrimmon, Minimal ideals in quadratic Jordan algebras, Proc. Amer. Math. Soc. 88 (4) (1983) 579–583], inspired by the problem posed in 1971 by Zhevlakov in the Dniester Notebook: Unsolved Problems in the Theory of Rings and Modules (second ed., 1976) (see Filippov et al. (1993) [5]).