  We study the problem of constructing a general hybrid quantum-classical
bracket from a partial classical limit of a full quantum bracket. Introducing a
hybrid composition product, we show that such a bracket is the commutator of
that product. From this we see that the hybrid bracket will obey the Jacobi
identity and the Leibniz rule provided the composition product is associative.
This suggests that the set of hybrid variables belonging to an associative
subalgebra with the composition product will have consistent quantum-classical
dynamics. This restricts the class of allowed quantum-classical interaction
Hamiltonians. Furthermore, we show that pure quantum or classical variables can
interact in a consistent framework, unaffected by no-go theorems in the
literature or the restrictions for hybrid variables. In the proposed scheme,
quantum backreaction appears as quantum-dependent terms in the classical
equations of motion.
