Commutativity equations and their trigonometric solutions

In the theory of Frobenius manifolds and Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations, one normally assumes that Frobenius algebras associated with a solution F have an identity e. Equivalently, the corresponding flat metric can be expressed as a linear combination of the matrices of the third-order derivatives of the pre-potential function F. We show that under certain non-degeneracy conditions, this assumption can be omitted, that is, the identity field e exists automatically without further assumptions. We also study trigonometric solutions F determined by a finite collection of vectors with multiplicities, and we give an explicit formula for the field e for all the known such solutions. The corresponding collections of vectors are given by the non-simply laced root systems or are related to their projections to the intersection of mirrors.