I am reading a proof on p.51 of Robert Steinberg in his book "Endomorphisms of Algebraic Groups" and I am having a bit of difficulty understanding one point in the proof.


The setting is as follows. Consider an algebraic group $G$ over an algebraically closed field of arbitrary characteristic. We say that an automorphism $\sigma:G\to G$ is *semisimple* if there exists an embedding $G\hookrightarrow G'$ such that $\sigma$ is realised by conjugation in $G'$ by a semisimple element. The theorem is:



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> If $G$ is solvable and $\sigma:G\to G$ is a semisimple automorphism then there exists a maximal torus $T\subseteq G$ such that $\sigma(T) = T$.
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I shall paraphrase the proof on pp.51-52 of Steinberg. Assume $G$ is connected. We choose at first an arbitrary maximal torus $T\subseteq G$, and let $U$ be the normal subgroup of $G$ of all unipotents; then we have $G = T\ltimes U$. We embed $G$ in a larger group so that $\sigma:G\to G$ is realised by conjugation by a semisimple element $s$. Then $sTs^{-1}$ and $T$ are maximal tori of $G$, so we can write $T = usTs^{-1}u^{-1}$ where $u\in G$ and by our decomposition, we can assume $u\in U$.


It then suffices to show that $us$ is actually conjugate to $s$: i.e. $us = xsx^{-1}$ for some $x\in G$, for then $\sigma$ will fix $x^{-1}Tx$. 


Then Steinberg says: *"and on replacing $us$ by its semisimple part we may assume it to be semisimple."* After this, he proceeds to show that if $us$ is semisimple, then it indeed is conjugate to $s$, which completes the proof.



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> Question: why can we assume that $us$ is semisimple? This *would* follow if the semisimple part could be written as $u's$ where $u' \in U$, but I cannot see why this is true at the moment.
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Thanks!



Confusing Point in Proof: Semisimple Automorphism Fixes Torus
Consider yourself fortunate to be having only "a bit of difficulty" in reading Steinberg's concise proofs. His style typically involves numerous reductions to get to the essential point, along with sentences crammed with dense arguments. Though he is almost always right on target, it can be frustrating to unpack such proofs (as I learned the hard way in graduate school when giving seminar talks on two of his papers from 1963 and 1965). 


Here you are looking at his 7.6 in the well-filled AMS memoir published in 1968 but mostly developed earlier. (I'm looking at the typeset version in his collected papers, where the page numbers are different.) It's helpful to add some notation, which he wisely avoids as far as possible. The essential situation involves a connected solvable algebraic group $G= T \ltimes U$ with $T$ any maximal torus and $U$ the unipotent radical. A basic fact is that all maximal tori of $G$ are conjugate under elements of $U$ (a step to the general conjugacy theorem). We are given some embedding of $G$ into a possibly larger algebraic group $G'$ and a "semisimple" automorphism $\sigma$ of $G$ coming from conjugation by a semisimple $s \in G'$. The problem is to show that $\sigma$ stabilizes some maximal torus of $G$. 


Now $T$ itself might not be $\sigma$-stable, but $us T s^{-1} u^{-1} = T$ for some $u \in U$. Thus $us \in N\_{G'}(T)$, where its Jordan decomposition in $G'$ lives. Note too that $us \in N\_{G'}(G)$ since each factor does; thus the Jordan parts of $us$ stabilize $G$ and also $T$. In particular, it is harmless now to replace $us$ by its semisimple part in $G'$ (i.e., assume it is semisimple). This is Steinberg's reduction.


After this reduction, the remaining point is to show that $us$ is actually conjugate in $G'$ to $s$. Moreover, the conjugating element stabilizes $G$ and takes $T$ to a new maximal torus which will be $\sigma$-stable as desired. Here again the argument has to be reduced, now to the case when $U$ is commutative. Etc. 


P.S. Steinberg's last steps are also compressed (using the commutativity of $U$ in a devious way), but his reference to page 4-14 in Expose 4 of the 1956-58 Chevalley seminar 
[here](http://www.numdam.org/numdam-bin/feuilleter?id=SCC_1956-1958__1_) is again elementary. 


As Steinberg points out, a different proof (which actually uses more of the Chevalley structure theory) was given by David J. Winter in Corollary 4
[here](http://www.ams.org/journals/bull/1966-72-04/S0002-9904-1966-11563-X/). 


