[This question arises from trying to understand  an incompletely formulated earlier question
<a href="https://mathoverflow.net/questions/111341/">here</a>.]

Let $G$ be a semisimple algebraic group over an algebraically closed field $K$ of 
characteristic $p \geq 0$, with some fixed maximal torus $T$ of dimension $r$ (the rank of $G$).
The *adjoint quotient* $\pi:G \rightarrow X$ involves an irreducible affine variety $X$ of dimension
$r$ associated to the algebra of invariants $K[G]^G$ (Chevalley).    Here $X$ identifies with the orbit space of $T$ under the action of the Weyl group.

Each fiber of $\pi$ is the union of one or more conjugacy classes, but only the semisimple classes are actually closed.   Call $\sigma:X \rightarrow G$ a *section* of $\pi$ if $\pi \sigma = \text{id}_X$.   Then $S:= \sigma(X)$ is an example of a *cross-section*: a closed irreducible subset of $G$ meeting each fiber of $\pi$ in a single point.    Call cross-sections $S$ and $S'$ "conjugate" if there is a morphism $\gamma:X  \rightarrow G$ (possibly constant) such that $\gamma(s) s \gamma(s)^{-1} \in S'$ for each $s \in S$.  .

> General problem: Classify all cross-sections of $\pi$, up to conjugacy.

In his 1965 *Publ. Math. IHES* paper _Regular elements of semi-simple algebraic groups_ (<a href="http://www.numdam.org/item/PMIHES_1965__25__49_0/">Numdam</a>),
Steinberg assumed that $G$ is *simply connected* and constructed an explicit section $\sigma$ 
of $\pi$.    Its image $S \subset G$ in fact consists entirely of regular points; each fiber
is the closure of a regular class and contains a unique (closed) semisimple class.    In the simply
connected case, the affine variety $X$ can be identified naturally with $\mathbb{A}^r$, via the
characters of fundamental representations (for positive roots relative  to $T$.)

*Example*: When $G=\mathrm{SL}_n(K)$, Steinberg's cross-section involves a version of the
classical rational canonical form.   If $n=3$ one gets essentially the companion matrices of
the characteristic (=minimal) polynomials $x^3 - a x^2 + bx -1$:

$$ \begin{pmatrix}
a & b & 1 \\
1 & 0 & 0 \\
0 & 1 & 0
\end{pmatrix} $$

In a 2003 AMS conference proceedings,  Friedman and Morgan, in <a href="https://arxiv.org/abs/math/0209053"><i>Automorphism sheaves, spectral covers, and the Kostant and Steinberg sections</i></a>, work exclusively over $\mathbb{C}$.  Their main theorem shows that (for $G$
still simply connected) all cross-sections are conjugate to the Steinberg cross-section.  But they use methods of complex algebraic geometry which go well beyond Steinberg's relatively elementary group-theoretic framework.

>(!)  If $p=0$, is there a more direct proof of the Friedman-Morgan theorem in the algebraic group setting, valid over
any algebraically closed field of characteristic 0?

The $p=0$ assumption is actually necessary here, as shown more recently by Vladimir L.
Popov in *Cross-sections, quotients, and representation rings of semisimple algebraic groups* in Transformtion Groups 16 (2011),  827-856.    Popov works over our field $K$, where if $p=0$ he 
observes that all cross-sections arise from sections of $\pi$.     (But if the natural map from a simply connected covering group to $G$ fails to be bijective, there are no cross-sections, so he studied instead "rational" sections over a dense open subset of $X$ for any $p$.)

It's worse when $p>0$.   By replacing the 0 in the center of the above matrix by $a^{p^d}-a$ for $d>0$,
Popov gets cross-sections which don't come from sections and are not all conjugate (for separability
reasons).   

>  (2) If $p>0$,  can one classify the cross-sections up to conjugacy?   (Is the Steinberg section somehow special?)