Necessity of surjectivity in commutator-ideal splitting for Jordan homomorphisms

Ascertain whether the surjectivity assumption in the following statement is necessary: if A is a unital associative algebra with commutator ideal K(A)=A, then for every Jordan homomorphism φ: A → B there exists a central idempotent e in B such that x ↦ eφ(x) is a homomorphism and x ↦ (1−e)φ(x) is an antihomomorphism.

Background

The paper proves that for unital algebras A satisfying K(A)=A, every Jordan epimorphism φ: A → B splits via a central idempotent into a homomorphism and an antihomomorphism. This provides a strong structural insight when the image of φ is an associative algebra.

It remains unclear whether the epimorphism (surjectivity onto its image) requirement is essential. Removing this assumption would significantly strengthen the commutator-ideal approach to Jordan homomorphisms and broaden applicability beyond epimorphic cases.