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

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
1007.2777. Some technical steps rely on the forthcoming book *Pseudo-reductive groups* (Cambridge, 2010) by Conrad-Gabber-Prasad.