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 is needed. Possibly the known counterexamples using Witt vectors suggest in some way all possible counterexamples? (Or is the question hopeless to resolve completely?)