A linear subspace M of a Jordan algebra J is said to be a Lie triple ideal of J if [M,J,J] ⊆ M, where [·,·,·] denotes the associator. We show that every Lie triple ideal M of a nondegenerate Jordan algebra J is either contained in the center of J or contains the nonzero Lie triple ideal [U,J,J], where U is the ideal of J generated by [M,M,M].
Let H be a Jordan algebra, let J be a prime nondegenerate Jordan algebra with extended centroid C and unital central closure Ĵ, and let Φ: H → J be a Lie triple epimorphism (i.e. a linear surjection preserving associators). Assume that deg(J) ≥ 12. Then we show that there exist a homomorphism Ψ: H → Ĵ and a linear map τ: H → C satisfying τ([H,H,H]) = 0 such that either Φ = Ψ + τ or Φ = -Ψ + τ.
Using the preceding results we show that the separating space of a Lie triple epimorphism between Jordan-Banach algebras H and J lies in the center modulo the radical of J.