  We describe relations between maximal subfields in a division ring and in its
rational extensions. More precisely, we prove that properties such as being
Galois or purely inseparable over the centre generically carry over from one to
another. We provide an application to enveloping skewfields in positive
characteristics. Namely, there always exist two maximal subfields of the
enveloping skewfield of a solvable Lie algebra, such that one is Galois and the
second purely inseparable of exponent 1 over the centre. This extends results
of Schue in the restricted case. Along the way we provide a description of the
enveloping algebra of the p-envelope of a Lie algebra as a polynomial extension
of the smaller enveloping algebra.
