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 http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1102991730. 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
1967 Tits responded to my inquiry with a letter outlining the behavior of the
$SL_2$ 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 1007.2777. Some technical steps rely on the forthcoming book Pseudo-reductive groups (Cambridge, 2010) by Conrad-Gabber-Prasad.