TITLE: Mostow's theorem on algebraic groups
QUESTION [7 upvotes]: In his classical 1956 paper 
Fully reducible subgroups of algebraic groups
Mostow proves the following theorem:

Theorem 7.1.
  Let $G$ be an algebraic group over a field $K$ of characteristic 0, 
  $\mathfrak{N}$ the set of nilpotent elements in the radical of its Lie algebra,
  and $N$ the connected algebraic group with Lie algebra $\mathfrak{N}$.
  Then $G=M\cdot N$ (semi-direct) with $M$ a maximal fully reducible group.
  Furthermore any two maximal fully reducible subgroups of $G$
  are conjugate under an inner automorphism from $\mathfrak{N}$.

This paper was written long before the Three Books on linear algebraic groups were published, 
and I do not fully understand Mostow's terminology.
Is "a fully reducible group" the same as "an algebraic group with reductive identity component" (in characteristic 0)?
Does the following result follow from Mostow's theorem?

Theorem? 
  Let $G$ be a linear algebraic group, not necessarily connected, over a field $K$ of characteristic 0.
  Let $G^0$ denote the identity component of $G$, and let $N$ denote the unipotent radical of $G^0$.
  Then the extension
  $$
1\to N\to G\to G/N\to 1
$$
  splits, i.e. $G=M\cdot N$ (a semidirect product),
  where $M\subset G$ is a $K$-subgroup of $G$ isomorphic to $G/N$.

Are there modern proofs of Mostow's theorem? References will be appreciated.

REPLY [4 votes]: Mostow's theorem actually has a relatively modern textbook treatment, by his early collaborator Gerhard Hochschild: see Theorem 4.3 in VIII.4 of his book Basic Theory of Algebraic Groups and Lie Algebras (GTM 125, Springer, 1981).   Hochschild followed the original attempt by Chevalley (in volume 2 of his Theorie des groupes de Lie) to transfer to affine algebraic groups over arbitrary fields the main ideas of Lie group theory.   Though this approach eventually becomes unsatisfactory in prime characteristic (especially over fields which are not algebraically closed), as Chevalley recognized, it does work fairly well in characteristic 0.   
In particular, Hochschild is able to reformulate in somewhat more modern language some of Mostow's insights into Levi decomposition.   For algebraic groups, the unipotent radical supplants the traditional solvable radical in Lie group theory due to the nicer description of solvable algebraic groups and the good behavior of tori.   It's worth taking a look at Hochschild's proof of Mostow's theorem and related results, since he provides a fairly elementary unified framework that still makes sense to people working today on more sophisticated problems in algebraic geometry.    Any way the proof is approached, it definitely has prerequisites, as the version here by grp illustrates.   It's hard to say what version is "easiest", since that depends heavily on the reader's background.
In any case, Hochschild also points out why the restriction to characteristic 0 is essential.   The notion of "reductive" algebraic group is meaningful in any characteristic, but "fully reducible" (or "linearly reductive" in Hochschild's more current language) applies only to tori among the connected algebraic groups in prime characteristic (Nagata).   Indeed, there are serious problems with the notion of Levi decomposition (and conjugacy of Levi factors) in the latter case, which George McNinch has recently been studying.    Fortunately Mumford's geometric invariant theory survived, due to the proof by Haboush of his conjecture that reductive groups are at least "geometrically reductive" in all characteristics.   But Lie groups don't serve well enough as a model for linear algebraic groups in general.