The final Moufang variety: FRUTE loops

FRUTE loops are loops that satisfy the identity $(x\cdot xy)z=(y\cdot zx)x$. We show that locally finite FRUTE loops are precisely the products $O\times H$, where $O$ is a commutative Moufang loop in which all elements are of odd order, and $H$ is a $2$-group such that the derived subloop $H'$ is of exponent two and $H'\le Z(H)$.