We consider commutativity equations $F_i F_j =F_j F_i$ for a function $F(x^1,
    \dots, x^N),$ where $F_i$ is a matrix of the third order derivatives $F_{ikl}$.
    We show that under certain non-degeneracy conditions a solution $F$ satisfies
    the WDVV equations. Equivalently, the corresponding family of Frobenius
    algebras has the identity field $e$. We also study trigonometric solutions $F$
    determined by a finite collection of vectors with multiplicities, and we give
    an explicit formula for $e$ for all the known such solutions. The corresponding
    collections of vectors are given by non-simply laced root systems or are
    related to their projections to the intersection of mirrors.

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