  We prove that the universal enveloping algebra of a Poisson-Ore extension is
a length two iterated Ore extension of the original universal enveloping
algebra. As consequences, we observe certain ring-theoretic invariants of the
universal enveloping algebras that are preserved under iterated Poisson-Ore
extensions. We apply our results to iterated quadratic Poisson algebras arising
from semiclassical limits of quantized coordinate rings and a family of graded
Poisson algebras of Poisson structures of rank at most two.
