  We show that a relation among minimal non-faces of a fillable complex $K$
yields an identity of iterated (higher) Whitehead products in a polyhedral
product over $K$. In particular, for the $(n-1)$-skeleton of a simplicial
$n$-sphere, we always have such an identity, and for the $(n-1)$-skeleton of a
$(n+1)$-simplex, the identity is the Jacobi identity of Whitehead products
($n=1$) and Hardie's identity of higher Whitehead products ($n\ge 2$).
