Existence of idempotent elements in arbitrary evolution algebras

Determine whether every evolution algebra over an arbitrary field contains a nonzero idempotent element, i.e., ascertain the existence of u ∈ E with u^2 = u ≠ 0 for an arbitrary evolution algebra E.

Background

The paper analyzes idempotents in evolution algebras via a polynomial system linking structure constants to the idempotent condition. For regular (perfect) complex evolution algebras, the authors prove the existence of idempotents.

However, the general existence question for arbitrary evolution algebras is unresolved and explicitly identified as open, referencing Mukhamedov and Qaralleh (2023).