Classical theorems attributed to Levi, Mal'cev, Harish-Chandra for a finite
dimensional Lie algebra over a field of characteristic 0 state that it has a Levi decomposition (semisimple subalgebra plus solvable radical) and that all such semisimple subalgebras (Levi factors) are conjugate in a strong sense: see Jacobson, *Lie Algebras*, III.9, for example.    This carries over to connected linear algebraic
groups, but in prime characteristic there are counterexamples going back perhaps
to Chevalley that involve familiar group schemes like $SL_2$ over rings of Witt
vectors.    Recent posts here have somewhat ignored that difficulty, having just characteristic 0 in mind.   Borel and Tits redefined "Levi factor" to be a reductive complement to the unipotent radical, which is makes no real difference in characteristic 0 but allows them to concentrate on positive answers for parabolic subgroups of reductive groups in general.    Other familiar subgroups of reductive groups like the identity component of the centralizer of a unipotent element require much more subtle treatment, as in work of George McNinch.

Whether or not the characteristic $p$ question is important, it has remained
open for many decades (say over an algebraically closed field).   I gave up after one forgettable paper (Pacific J. Math. 23, 1967).   The problem is still
easy to state: 

>Are there effective necessary or sufficient conditions for existence or uniqueness of Levi factors in a connected linear algebraic group over an algebraically closed field of prime characteristic?

It's clear that a scheme-theoretic viewpoint may be needed.   Possibly the known
counterexamples using Witt vectors suggest in some way all possible counterexamples?    (Or is the question hopeless to resolve completely?)

EDIT: For online access to my 1967 paper, via Project Euclid, see
[Link](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-23/issue-3/Existence-of-Levi-factors-in-certain-algebraic-groups/pjm/1102991730.full).   Here
Chevalley's counterexample is mentioned only in the abstract, but in remarks
later on it is noted that Borel-Tits (III.15) gave an example involving two
Levi subgroups which fail to be conjugate; see NUMDAM link to PDF version of
Publ. Math. IHES 27 (1965) at http://www.numdam.org:80/?lang=en

In April
1967 Tits responded to my inquiry with a letter outlining the behavior of the
group scheme $SL_2$ over the ring of Witt vectors of length 2, which gives a
6-dimensional algebraic group over the underlying field with unipotent radical of dimension 3 but no Levi factor.  He remarked that he got this counterexample from P. Roquette but had also been told about Chevalley's counterexample.

ADDED: The question as formulated probably doesn't have a neat answer, but meanwhile George McNinch has delved much deeper (over more general fields) in his new arXiv preprint
<a href="http://front.math.ucdavis.edu/1007.2777" rel="nofollow">1007.2777</a>.  Some technical steps rely on the forthcoming book  *Pseudo-reductive groups* (Cambridge, 2010) by Conrad-Gabber-Prasad.