Normal structure of isotropic reductive groups over rings
Anastasia Stavrova Alexei Stepanov
math.GR math.KT


The paper studies the lattice of subgroups of an isotropic reductive
group G(R) over a commutative ring R, normalized by the elementary subgroup E(R).
We prove the sandwich classification
theorem for this lattice under the assumptions that the isotropic rank of G is at least 2
and the structure constants are invertible in R. The theorem asserts that the lattice splits into a disjoint union
of sublattices (sandwiches) E(R, )≤…≤ C(R, ) parametrized by the ideals 
of R, where E(R, ) denotes the relative elementary subgroup and C(R,) is
the inverse image of the center under the natural homomorphism G(R)→ G(R/).
The main ingredients of the proof are the “level computation” by the first author and
the generic element method developed by the second author.


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    Daan Frenkel
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§ INTRODUCTION


Let G be a reductive group scheme over a unital commutative ring R in the sense of <cit.>.
The famous result of J. Tits <cit.>
establishes that if R is a field, and G has no normal closed connected smooth R-subgroups, then its
group of R-points G(R) is very
close to being simple as an abstract group (except in a few cases where R=𝔽_2 or 𝔽_3).
Namely, G(R) contains a “large” normal subgroup G(R)^+ whose central
quotient is simple. If R is a finite field, the corresponding simple group G(R)^+/(G(R)^+) is a finite
simple group of Lie type, and in fact such groups constitute the largest family  in the classification of finite simple
groups <cit.>.

If R is no longer a field, then every proper ideal  of R determines a normal subgroup G(R,) of G(R) called
the principal congruence subgroup of level ; this subgroup is the kernel
of the natural homomorphism ρ_:G(R)→ G(R/). Thus G(R) is no longer simple, and its lattice of normal subgroups
is at least as rich as the lattice of ideals in R. H. Bass <cit.> established that,
at least if G=_r and r>max( R,2), every normal subgroup in G(R) is sandwiched
between the “small” elementary congruence subgroup E(R,) and the “large” full congruence subgroup
C(R,), to be defined below (the discription for _2 is very different <cit.>). This revelation was instrumental in the proof of the congruence subgroup problem
for _n (n≥ 3) and _2n (n≥ 2) by J. Mennicke, H. Bass, M. Lazard, J. Milnor, and J.-P.
Serre <cit.>. Further work on the congruence subgroup problem for Chevalley groups <cit.>,
as well as progress in algebraic K-theory, inspired extensions of
this normal structure result to split reductive groups over general commutative rings.
In particular, L. Vaserstein <cit.>
proved the standard normal sructure of Chevalley groups G of rank ≥ 2
over commutative rings with the only condition that the structure constants of G are invertible in R.
The congruence subgroup problem for absolutely almost simple simply connected algebraic groups of isotropic rank ≥ 2
over global fields was solved by
M. S. Raghunathan <cit.>, who obtained, as an important intermediate
step, a description of normal subgroups in arithmetic subgroups of these algebraic groups.
However, the description of normal subgroups for isotropic reductive groups over general rings has been known only in a few
distinct cases, such as orthogonal and hyperbolic unitary groups <cit.>.
The goal of the present paper is to complete this line of thought by proving that
the standard description of normal subgroups holds for all reductive groups G
of isotropic rank ≥ 2, under the same necessary assumptions on R as for split groups.










To be more precise, we say that G has isotropic rank ≥ n, if every
semisimple normal R-subgroup of G contains an n-dimensional split R-torus
(𝔾_m,R)^n. If the isotropic rank is ≥ 1, then G contains a pair of opposite
strictly proper
parabolic R-subgroups
P^± <cit.>, and one defines the elementary subgroup E_P(R) as the subgroup of G(R)
generated by U_P^+(R) and U_P^-(R), where U_P^± denotes the unipotent radical  of P^±.
If, moreover, the isotropic rank of G is ≥ 2, the main result of <cit.> implies that E(R)=E_P(R)
is independent of the choice of P^± and is normal in G(R) (see  <ref> for the details).

If G=_n, n≥3, then E(R) is the usual elementary subgroup
generated by the elementary transvections e+te_ij, 1≤ i≠ j≤ n, t∈ R
(here e denotes the identity matrix whereas e_ij is the matrix with 1 at position (i,j)
and zeros elsewhere).
If R is a field, then E(R)=G(R)^+ is the above-mentioned group of J. Tits.










Assume that the isotropic rank of G is ≥ 2. For any ideal  of R, let U_P^±()
be the kernel of the homomorphism ρ_ restricted to U_P^±(R). Denote by
E_P(R,) the normal closure of the subgroup generated by U_P^+() and U_P^-() in E(R).
The full congruence subgroup C(R,) is the full preimage of the center of G(R/) under ρ_.

For any maximal ideal  of R, denote by R/ the algebraic closure of the field
R/. The group G_R/ is a reductive algebraic group in the usual sense <cit.>, and thus
has a root system Φ in the sense of Bourbaki <cit.>.
Note that G_R/ has no closed connected normal smooth R-subgroups if and only if Φ is irreducible.
The structure constants of Φ are, by definition, the integers ± 1,
together with ± 2, if Φ
is of type B_n,C_n,F_4, or ± 2,± 3, if Φ is of type G_2.

The main result of the present paper is the following theorem.


Let G be a reductive group scheme over a ring R such that its isotropic rank is ≥ 2.
Suppose that for every maximal ideal  of R the root system of
G_R/ is irreducible, and its structure constants are invertible in R.
Let P be a proper parabolic subgroup of G.
Then
(enumi)


  * For any ideal  of R one has E_P(R,)=[G(R,),E(R)]. In particular,
E_P(R,)=E(R,) is independent of the choice of a parabolic R-subgroup P.


  * For any subgroup H≤ G(R)
normalized by E(R), there exists a unique ideal  in R such that

    E(R, q)≤ H≤ C(R,).




Since groups of points of reductive group schemes include, in particular, the linear matrix groups
_n(R), _n(R), _2n(R), as well as spinor and special orthogonal groups (q)(R), (q)(R), where
q is a non-degenerate quadratic R-form, the study of their normal structure has a very long history.
Thus, it would take a separate survey paper to describe it in full, and below we only list
a few milestone results.



  * Simplicity of the groups PSL_n(F) was proved by Camille Jordan around 1870 (for prime fields)
and in early 1900 by Leonard Dickson (for all finite fields).

  * A similar result for the GL_n over a skew-field was obtained by J. Dieudonne in 1943. In <cit.>
he established the simplicity of split groups over arbitrary fields. The first uniform proof of their simplicity was given by
C. Chevalley <cit.>.

  * The problem over rings distinct from fields was first approached by J. Brenner <cit.> and
later by W. Klingenberg <cit.>. They studied split groups of classical types
over quotients of ℤ and general local rings respectively.


  * As mentioned above, J. Tits established the simplicity for isotropic reductive groups over fields in <cit.>.

  * For the general linear group over an arbitrary ring,
the normal structure theorem was proved by H. Bass <cit.>
under the stable range condition. The result was generalized to other quasi-split classical
groups by H. Bass himself <cit.> and A. Bak <cit.>.


  * H. Bass, M. Lazard, J.-P. Serre, and J. Milnor <cit.>
elucidated the normal structure of _n and _2n over
rings of integers of algebraic number fields, and stated the congruence subgroup problem.
It was solved for _2 by J.-P. Serre <cit.> and for all split groups by H. Matsumoto <cit.>.


  * The stable range condition for GL_n was removed by
J. Wilson <cit.> (n≥4) and I. Golubchik <cit.> (n≥3).
Using the ideas of H. Bass and Suslin's theorem on the normality of the elementary group <cit.>,
Z. Borewich and N. Vavilov <cit.> gave a simpler prove of the Wilson–Golubchik theorem.


  * In 1974 E. Abe and K. Suzuki proved the normal structure theorem for all Chevalley groups over local
rings <cit.>.



  * In 1976 M.S. Raghunathan established the congruence subgroup problem for groups of isotropic rank ≥ 2 over
a global field <cit.>. In 1986 he improved this result, weakening the isotropy conditions <cit.>.


  * In 1979 G. Margulis <cit.> proved his celebrated theorem on lattices in isotropic groups over local
fields. His work was based on earlier work of V. P. Platonov <cit.>.


  * In 1980 R. Bix established the normal structure theorem for isotropic groups of type 1E_6,2^28 over
local rings <cit.>.



  * L. Vaserstein in <cit.> combined the results of H. Bass, J. Wilson and I. Golubchik
proving the standard normal structure of the GL_n under a local stable rank condition.


  * After the result of G. Taddei on normality of the elementary group in a Chevalley group
L. Vaserstein <cit.> proved the standard normal sructure of Chevalley groups of rank ≥ 2
over commutative rings
provided that the structure constants were invertible.
The latter condition was removed by E. Abe in <cit.>.


  * In 1988 L. Vaserstein proved the normal structure theorem for isotropic orthogonal groups over commutative
rings <cit.>.


  * The result of E. Abe has three exceptions where the elementary group is not perfect:
types C_2 and G_2 if the gound ring has a residue field of 2 elements, and type A_1.
All the exceptions were considered by D. Costa and G. Keller in a series of papers <cit.>.
Of course, for type A_1 the ground ring must be low-dimensional with many units, as the
 description of the normal structure of SL_2(ℤ) seems to be an unrealistic problem.


  * In 2010 Zuhong Zhang <cit.> established the normal structure theorem for even hyperbolic
unitary groups over a commutative form ring with invertible 2. The latter assumption was removed by Hong You <cit.>.
A shorter proof was obtained recently by R. Preusser <cit.>.


The main ingredients of the proof of Theorem <ref> is a level computation made by the first author
in <cit.> and the generic element method developed by the second author in <cit.>.
Our proof basically follows the plan developed by the second author in <cit.>.







































































§ BASIC NOTATION AND CONVENTIONS

Let x,y,z be elements of an abstract group H. Denote by yx=yxy^-1 the left
conjugate to x by y. Sometimes we use also the right conjugate x^y=y^-1xy to
x by y. The commutator xyx^-1y^-1 is denoted by [x,y].
In the sequel we frequently use the following commutator identity, which can be easily verified
by a straightforward calculation.



    [zy,x]^z= yx·(x^-1)^z=[y,x]·[x,z^-1]



Let S be a subset of H. By ⟨ S⟩ we denote the subgroup
spanned by S. For  subsets X and Y of H
[X,Y] stands for the subgroup of H generated by all the commutators
[x,y], x∈ X, y∈ Y. Note that if Y is a subgroup, then [X,Y] is normalized by Y
(apply (<ref>) to the case, where x∈ X and z and zy are arbitrary elements of Y).
If both X and Y are subgroups, then we denote by X^Y the subgroup of H
generated by x^y for all x∈ X and y∈ Y. In other words, X^Y is the smallest subgroup
containing X and normalized by Y.
The centralizer of a subgroup H'≤ H is denoted by _H(H') and
we write (H)=_H(H) for the center of H.

All rings and algebras are assumed to be commutative and to contain a unit.
All homomorphisms preserve unit elements.
The multiplicative group of a ring R is denoted by R^×.
As usual, Spec R denotes the prime spectrum of R.
For ∈Spec R denote by R_=(R)^-1R
the localization of R at p and by k() the residue
field R_/ R_. The algebraic closure of k() is denoted by k().

Let s∈ R. The principal localization at the element s
(i. e. the localization at the multiplicative subset generated by s)
is denoted by R_s. The localization homomorphism is denoted by λ_ p
or λ_s respectively.

Throughout the paper K is a ring, G stands for a reductive group scheme over K in the sense of <cit.>,
and R denotes a K-algebra, unless explicitly stated otherwise. Sometimes we replace G by G_R
and consider it as a group scheme over R. It makes no harm as we deal with
the group of points G(R)=G_R(R).

For any ideal  of R we denote
by ρ_:R→ R/ the reduction homomorphism, and, by abuse of notation, the induced homomorphism G(R)→ G(R/).
The principal congruence subgroup G(R, q) is the kernel of ρ_ q:G(R)→ G(R/ q),
whereas the full congruence subgroup C(R,) is the inverse image of the center (G(R/)) of
G(R/ q) under this homomorphism.

An ideal  of R is called splitting if R/ embeds into R such that the composition
R/↣ R↠ R/ is identity. In other words,  is a splitting ideal
if R=R'⊕ as additive groups, where R' is a subring of R.




§ GENERIC ELEMENT TECHNIQUES

Let G be an affine smooth finitely presented group scheme over K.

Denote by A=K[G] the affine algebra of the scheme G.
By the definition of an affine scheme, an element h∈ G(R) can be identified with a
homomorphism h:A→ R.
We always do this identification, i. e. we always view elements
of the group of points G(R) of the scheme G over a K-algebra R as homomorphisms
from A to R.[Thus, for f∈ A we write h(f) instead of f(h), as one writes
considering the affine algebra as a set of functions.]
Denote by g∈ G(A) the generic element of the scheme G, i. e.
the identity map id_A:A→ A. An element h∈ G(R) induces the homomorphism
G(h):G(A)→ G(R) by the rule G(h)(a)=h∘ a for all a∈ G(A).
It follows that the image of g under the action of G(h) is equal to h.

For a ring homomorphism :R→ R' we usually  denote the induced group homomorphism
G():G(R)→ G(R') again by .
This cannot lead to a confusion as one always can determine the meaning of 
by the argument type of this homomorphism.
In view of this agreement we have h(g)=h∘id_A=h.
If R' is an R-algebra, then sometimes we identify elements of G(R) with their canonical
images in G(R').

Recall that the fundamental ideal I of A is the kernel of the counit map e_K:A→ K, where
e_K∈ G(K) is the identity element of this abstract group.
The notation A, I, and g introduced above is kept till the end of the present section.

The following characterization
of the principal congruence subgroup was observed in <cit.>, see the paragraph before Lemma 2.1.


An element h of G(R) belongs to the principal congruence subgroup G(R,) if and only
if h(I)⊆.



Denote by ρ_:G(R)→ G(R/) the natural homomorphism.
It is easy to see that h∈ G(R,) iff the following diagram commutes.

    A  @>h>>   R   
    
    @Ve_KVV  @VVρ_V
    
    K  @>>>    R/

And the latter is obviously equivalent to saying that h(I) vanishes modulo ,
i. e. h(I)⊆.


Since A is a finitely presented K-algebra, it is a quotient of a polynomial ring in
finitely many variables by a finitely generated ideal. Let K' be a -subalgebra of K,
generated by all the coeffients of polynomials that generate this ideal. Then A≅ A'⊗_K'K,
were A' is a finitely generated algebra over a Noetherian ring K'. Thus, there exists
an affine smooth finitely presented group scheme G over a Noetherian ring K' such that G= G_K.
In the present paper we prove results about the abstract group G(R) for a K-algebra R.
Therefore, without loss of generality we may assume that K=K', and hence A is a
Noetherian ring.

An advantage of the Noetherian property of a ring in this context is the following
property <cit.>.


Let R be a Noetherian ring and s∈ R. There exists m∈ such that the restriction
of the localization homomorphism λ_s:R→ R_s to the ideal s^mR is injective.


The next lemma is a version of clearing denominators.


Let G be an affine smooth finitely presented group scheme over a ring K,  R a Noetherian K-algebra,
m∈, and s∈ R.
Suppose that the natural map s^mR→ R_s is injective.
Given a∈ G(R_s) there exists k∈ such that [a,λ_s(b)]∈λ_s(G(R,s^mR))
for all b∈ G(R,s^kR).



Since A is finitely presented, there exists a finite
set J⊆ A that generates A as a K-algebra.
Since A=K⊕ I as a K-module, we may assume that J⊆ I.
Obviously, under this condition J spans I as an ideal.

Identifying elements of G(A) and G(R_s) with their canonical images in G(A⊗_K R_s),
consider the commutator c=[a,λ_s(g)]∈ G(A⊗_K R_s,I⊗_KR_s).
By Lemma <ref> the finite set c(J) is contained in I⊗_K R_s. Clearly, there exists l∈
such that each element of this set can be written as r/s^l for some r∈ I⊗_K R.
Take k=l+m. Take b∈ G(R,s^kR) and consider the composition
d=mult∘(b⊗id)∘ c=[a,λ_s(b)], where
mult:R⊗_KR_s→ R_s denotes the multiplication homomorphism. By Lemma <ref>
b(I)⊆ s^kR, hence the set d(J) consists of the elements of the form
mult∘(b⊗id)(r/s^l)=s^kt/s^l=s^mt for some t∈ R.
Since the restriction of the localization homomorphism λ_s to s^mR is injective
and J generates the K-algebra A, the homomorphism d factors through d':A→ R,
which means that d=λ_s(d'). Moreover, the natural choice of d' provides that
d'(J)⊆ s^mR. Since J generates the fundamental ideal I, we have
d'(I)⊆ s^mR and by Lemma <ref> d'∈ G(R,s^mR).





§ ELEMENTARY SUBGROUP OF AN ISOTROPIC REDUCTIVE GROUP

Let P be a parabolic subgroup of G in the sense of <cit.>.
Since the base SpecR is affine, the group P has a Levi subgroup L_P <cit.>.
There is a unique parabolic subgroup P^- in G which is opposite to P with respect to L_P,
that is P^-∩ P=L_P, cf. <cit.>.  We denote by U_P the unipotent
radical of P.

Note that if L'_P is another Levi subgroup of P,
then L'_P and L_P are conjugate by an element u∈ U_P(R) <cit.>.
Since our proofs in the present paper do not depend on a particular choice of L_P or P^-,
we do not pay attention to this choice.


The elementary subgroup E_P(R) corresponding to P is the subgroup of G(R)
generated as an abstract group by U_P(R) and U_P^-(R).



A parabolic subgroup P in G is called
strictly proper, if it intersects properly every normal semisimple subgroup of G.


The following theorem is the main result of <cit.>.


Assume that for every maximal ideal  of R every normal semisimple
subgroup of G_R_ contains (𝔾_m,R_)^2. Then the subgroup E_P(R) of G(R)=G_R(R)
is the same for any strictly proper parabolic R-subgroup P of G_R.
In particular, E_P(R) is normal in G(R).



Under the assumptions of Theorem <ref> we call E_P(R) the elementary subgroup of G(R) and denote it by E(R).


We also use the following theorems, which are the main results of <cit.> and <cit.>.
We denote by (G) the group scheme center of G in the sense of <cit.>, see also discussion
around Proposition 6.7 in <cit.>. The very definition of (G) implies that
(G)(R)≤(G(R)).


Under the hypothesis of Theorem <ref>, one has

    _G(R)(E(R))=(G)(R)=(G(R)).




Under the hypothesis of Theorem <ref>,
assume, moreover, that for any maximal ideal  of R, one has
k()≠𝔽_2 whenever the root system of G_k()
contains an irreducible component of type B_2=C_2 or G_2.
Then E(R) = [E(R),E(R)].


The following statement will be used twice to get rid of the center.


Let H be a subgroup of G(R), normalized by E(R). Under the assumptions of Theorem <ref>
[[H,E(R)],E(R)]=[H,E(R)].
In particular, if [H,E(R)]≤(G(R)), then H≤(G(R)).



By Theorem <ref> the group E(R) is perfect.
Using the Hall–Witt identity we get

    [E(R),H]=[[E(R),E(R)],H]≤[[H,E(R)],E(R)].

The inverse inclusion is obvious. The second assertion follows immediately from the first one and
Theorem <ref>.






§ ROOT SYSTEMS CORRESPONDING TO PARABOLIC SUBGROUPS


Let S=(𝔾_m,R)^N=Spec(R[x_1^± 1,…,x_N^± 1])
be a split N-dimensional torus over R. Recall that the character group
X^*(S)=Hom_R(S,𝔾_m,R) of S is canonically isomorphic to ^N.
If S acts R-linearly on an R-module V, this module has a natural ^N-grading

    V=⊕_λ∈ X^*(S)V_λ,

where

    V_λ={v∈ V| s· v=λ(s)v for any s∈ S(R)}.

Conversely, any ^N-graded R-module V can be provided with an S-action by the same rule.

Assume that S acts on G by R-group automorphisms.
The associated Lie algebra functor (G) then acquires
a ^N-grading compatible with the Lie algebra structure,

    (G)=⊕_λ∈ X^*(S)(G)_λ.


We will use the following version of <cit.>.


Let L=_G(S) be the subscheme of G fixed by S. Let
Ψ⊆ X^*(S) be an R-subsheaf of sets closed under addition of characters.
Then there exists a unique smooth connected closed subgroup U_Ψ of G
normalized by L and satisfying


    (U_Ψ)=⊕_λ∈Ψ(G)_λ.


Moreover,


  * if 0∈Ψ, then U_Ψ contains L;

  * if Ψ={0}, then U_Ψ=L;

  * if Ψ=-Ψ, then U_Ψ is reductive;

  * if Ψ∪(-Ψ)=X^*(S), then U_Ψ and U_-Ψ are two opposite parabolic subgroups of G
with the common Levi subgroup U_Ψ∩(-Ψ);

  * If 0∉Ψ, then U_Ψ is unipotent.




The statement immediately follows by faithfully flat descent from the standard facts about the subgroups of
split reductive groups proved in <cit.>; see the proof of <cit.>.



The sheaf of sets

    Φ(S,G)={λ∈ X^*(S){0} | (G)_λ≠ 0}

is called the system of relative roots of G with respect to S.



Choosing a linear ordering on the -space ⊗_ X^*(S)≅^n, one defines the subsets
of positive and negative relative roots Φ(S,G)^+ and Φ(S,G)^-, so that Φ(S,G) is a disjoint
union of Φ(S,G)^+, Φ(S,G)^-, and {0}. By Lemma <ref> the closed subgroups

    U_Φ(S,G)^+∪{0}=P,     U_Φ(S,G)^-∪{0}=P^-

are two opposite parabolic subgroups of G with the common Levi subgroup _G(S).
Thus, if a reductive group G over R admits a non-trivial action of a split torus,
then it has a proper parabolic subgroup. The converse is true Zariski-locally,
see Lemma <ref> below.


Let P be a parabolic subgroup scheme of G over R, and let L be a Levi subgroup of P.
By <cit.> the root system Φ of G_k(), ∈SpecR,
is locally constant in the Zariski topology on SpecR. The type of the root system of
L_k() is determined by a Dynkin subdiagram
of the Dynkin diagram of Φ, which is also constant Zariski-locally on SpecR
by <cit.>. In particular, if SpecR is connected,
all these data are constant on SpecR.


Assume that the root system Φ of G_k() has the same type for all ∈SpecR.
Then we call Φ the absolute root system of G.



Assume that R is connected. Let L̅ be the image of L under the natural
homomorphism G→ G^ad⊆Aut(G).
Let D be the Dynkin diagram of the absolute root system Φ of G.
We identify D with a set of simple roots of Φ such that
P_k() is a standard positive parabolic
subgroup with respect to D. Let J⊆ D
be the set of simple roots such that D J is the subdiagram of D corresponing to L_k().
Then there are a unique maximal split subtorus
S⊆(L̅) and a subgroup Γ≤Aut(D) such that J is invariant under Γ
and
for any ∈Spec R and any split maximal torus T⊆L̅_k()
the kernel of the natural surjection

    X^*(T)≅Φ X^*(S_k())≅Φ(S,G)

is generated by all roots α∈ D J,
and by all differences α-σ(α), α∈ J, σ∈Γ.


In <cit.>, we introduced a system of relative roots Φ_P with respect to a parabolic
subgroup P. This system Φ_P was defined
independently over each member SpecR=SpecR_i
of a suitable finite disjoint Zariski covering

    SpecR=∐_i=1^mSpec R_i,

such that over each R_i, 1≤ i≤ m, the root system Φ and the Dynkin diagram D of G is constant.
Namely, we considered the formal projection

    π_J,ΓΦ⟶Φ/⟨ D J; α-σ(α)|α∈ J, σ∈Γ⟩,

and set Φ_P=Φ_J,Γ=π_J,Γ(Φ){0}. The last claim of Lemma <ref>
allows to identify Φ_J,Γ and Φ(S,G) whenever Spec R is connected.


In the setting of Lemma <ref> we call Φ(S,G)≅Φ_J,Γ a system
of relative roots with respect to the parabolic subgroup P over R and denote it by Φ_P.


If R is a field or a local ring, and P is a minimal parabolic subgroup of G,
then Φ_P is nothing but the relative root system of G with respect to a maximal split subtorus
in the sense of <cit.> or, respectively, <cit.>.

We have also defined in <cit.> irreducible components of systems of relative roots,
the subsets of positive and negative
relative roots, simple relative roots, and the height of a root. These definitions are immediate analogs
of the ones for usual abstract root systems, so we do not reproduce them here.
The height of a root α∈Φ is deonted by htα.

We will need later the following two lemmas on relative roots.


Let Φ be a root system, and let Φ_J,Γ=π(Φ){0} be a relative root system with the canonical
projection π:Φ→Φ_J,Γ.
Let α,β∈Φ_J,Γ be two simple relative roots such that α+β∈Φ_J,Γ.
Then for any j≥ 1 such that jβ∈Φ_J,Γ one has α+jβ∈Φ_J,Γ.



The inclusion α+β∈Φ_J,Γ is equivalent to the
existence of two simple roots in π^-1(β) and π^-1(α) respectively that
are connected in the Dynkin diagram by a chain (possibly, empty)
of simple roots lying in π^-1(0). Then for any element μ∈π^-1(jβ) the set S_μ⊆Φ of simple
roots  occurring in its decomposition
is also connected to a simple root in π^-1(α) by a chain of simple roots lying in π^-1(0),
which allows to find a root ν∈π^-1(α) such that S_μ and S_ν
are disjoint but adjacent subsets of the Dynkin diagram of Φ, and hence μ+ν∈Φ. Then
π(μ+ν)=α+jβ∈Φ_J,Γ.



Let Φ be a root system with the scalar product (,) and a system of simple roots
Π, and let Φ_J,Γ=π(Φ){0} be a relative root system with the canonical
projection π:Φ→Φ_J,Γ. Let β∈π(Π)∩Φ_J,Γ be a simple relative root.
Then there is a proper parabolic subset Σ of Φ_J,Γ
that contains all α∈Φ_J,Γ such that
α+β∉Φ_J,Γ. If Φ is simply laced, then

    Σ   ={α∈Φ_J,Γ| (a,∑_b∈π^-1(β) b)≥ 0  a∈π^-1(α)}
       ={α∈Φ_J,Γ| (a,∑_b∈π^-1(β) b)≥ 0  a∈π^-1(α)}.




If Φ is not simply laced, then there is a simply laced root system Φ' with a system of simple roots Π'
and a projection π':Φ'→Φ
such that Φ=Φ'_Π',Γ'
for a suitable group of automorphisms Γ' of the Dynkin diagram of Φ'.
 Then
Φ_J,Γ=Φ'_π'^-1(J),Γ”, where Γ” is a group of automorphisms of the Dynkin diagram
of Φ generated by Γ' and the natural lifting of Γ. This means that we can assume that
Φ is simply laced from the start.

For any α∈Φ_J,Γ and any μ∈π^-1(α) by <cit.> the set π^-1(α)
is a union of roots μ'∈Φ such that σ(μ_J)=μ'_J for some σ∈Γ, where
μ_J denotes the shape of μ with respect to J in the sense of <cit.> or <cit.>.
By <cit.>
the Weyl group of Π J acts transitively on the set of roots of a fixed shape and length, hence
this Weyl group and Γ together act transitively on π^-1(α). Since all elements of these groups
are bijections that preserve scalar products and the set π^-1(β), we conclude that
(μ,∑_ν∈π^-1(β)ν) is the same for all μ∈π^-1(α).
Thus, the set Σ of (<ref>)
is well-defined, and it is clear that Σ∪(-Σ)=Φ_J,Γ.

Assume that
α_1,α_2∈Σ and α_1+α_2∈Φ_J,Γ. By <cit.> for every
μ∈π^-1(α_1+α_2)
there are μ_1∈π^-1(α_1) and μ_2∈π^-1(α_2) such that μ=μ_1+μ_2.
This shows that Σ is additively closed. By the same argument, if α+β∉Φ_J,Γ, then
(μ,ν)≥ 0 for all μ∈π^-1(α) and ν∈π^-1(β), and hence α∈Σ.

It remains to show that Σ is a proper subset of Φ_J,Γ. Let α∈π(Π){0} be
a simple relative root such that α+β∈Φ_J,Γ. Since α-β∉Φ_J,Γ,
we conclude that (μ,ν)≤ 0 for all μ∈π^-1(α) and ν∈π^-1(β).
On the other hand, since α+β∈Φ_J,Γ,
there are two simple roots  μ∈π^-1(α)∩Π and ν∈π^-1(β)∩Π which are connected
on the Dynkin diagram by a chain of roots in Π J. Then the scalar product of μ and (ν+all roots in this chain)
is negative. Hence α∉Σ.





§ RELATIVE ROOT SUBSCHEMES

For any finitely generated projective R-module V, we denote by W(V) the natural affine scheme
over R associated with V, see <cit.> or <cit.>.
Any morphism of R-schemes W(V_1)→ W(V_2)
is determined by an element f∈Sym^*(V_1^∨)⊗_R V_2, where Sym^* denotes the symmetric algebra,
and V_1^∨ denotes the dual module of V_1. If f∈Sym^d(V_1^∨)⊗_R V_2,
we say that the corresponding morphism is homogeneous of degree d.
By abuse of notation, we also write f:V_1→ V_2 and call it degree d
homogeneous polynomial map from V_1 to V_2. In this context, one has

    f(r v)=r^d f(v)

for any v∈ V_1 and r∈ R. Moreover, for any R-algebra A the induced map V_1⊗_RA→ V_2⊗_RA,
v⊗ a↦ f(v)⊗ a^d, also is denoted by f.

In this section P denotes a parabolic subgroup of G and L=L_P stands for its Levi subgroup.

<cit.>.
In the setting of Lemma <ref>, for any α∈Φ_P=Φ(S,G) there exists a closed
S-equivariant  embedding of R-schemes

    X_α W((G)_α)→ G,

satisfying the following condition.



  (*)
Let A be an R-algebra such that G_A is split with
respect to a maximal split A-torus T⊆ L_A. Let e_δ,
δ∈Φ, be a Chevalley basis of (G_A), adapted to T and P, and
x_δ𝔾_a→ G_A, δ∈Φ, be the associated
system of 1-parameter root subgroups
(e.g. x_δ=exp_δ of <cit.>).
Let

    π:Φ=Φ(T,G_A)→Φ_P∪{0}

be the natural projection.
Then for any
u=∑_δ∈π^-1(α)a_δ e_δ∈(G_R)_α

one has


    X_α(u)=
    (∏_δ∈π^-1(α)x_δ(a_δ))·∏_i≥ 2(∏_θ∈π^-1(iα)x_θ(p^i_θ(u))),


where every p^i_θ:(G_A)_α→ A is a homogeneous polynomial map of degree i,
and the products over δ and θ are taken in any fixed order.




The images of closed embeddings X_α, α∈Φ_P, satisfying the statement of Lemma <ref>,
are called relative root subschemes of G with respect to the parabolic subgroup P and will be denoted by
_α.


Relative root subschemes of G with respect to P, actually,
depend on the choice of a Levi subgroup L_P, but their essential properties stay the same,
so we usually omit L_P from the notation.

Set V_α=(G)_α for short.
We will use the following properties of relative root subschemes.

<cit.>
Let X_α, α∈Φ_P, be as in Lemma <ref>.
Then

(i) There exist degree i homogeneous polynomial maps q^i_α:V_α⊕ V_α→ V_iα,
i>1, such that for any R-algebra A and for any
v,w∈ V_α⊗_RA one has

    X_α(v)X_α(w)=X_α(v+w)∏_i>1X_iα(q^i_α(v,w)).


(ii) For any g∈ L(R), there exist degree i homogeneous polynomial maps
φ^i_g,α V_α→ V_iα, i≥ 1, such that for any R-algebra A and for any
v∈ V_α⊗_R A one has

    gX_α(v)g^-1=∏_i≥ 1X_iα(φ^i_g,α(v)).


(iii) (generalized Chevalley commutator formula) For any α,β∈Φ_P
such that mα≠ -kβ for all m,k≥ 1,
there exist polynomial maps

    N_α,β,i,j V_α× V_β→ V_iα+jβ, i,j>0,

homogeneous of degree i in the first variable and of degree j in the second
variable, such that for any R-algebra A and
for any u∈ V_α⊗_RA, v∈ V_β⊗_RA one has

    [X_α(u),X_β(v)]=∏_i,j>0X_iα+jβ(N_α,β,i,j(u,v))


(iv) For any subset Ψ⊆ X^*(S){0} that is closed under addition,
the morphism

    X_Ψ W( ⊕_α∈ΨV_α)→ U_Ψ,    
    (v_α)_α∈Ψ↦∏_α X_α(v_α),

where the product is taken in any fixed order,
is an isomorphism of schemes.


Apart from the above properties of relative root subschemes we will use the following Lemma,
which appeared first as <cit.> and in a slighly stronger form in <cit.>.
Note that in both cases the original statements erroneously claim that the image (N_α,β,1,1)
(respectively, the sum of images in (2) ) equals V_α+β, while in reality
the respective proofs establish only that it generates V_α+γ
as an R-module, and hence as an abelian group. The correct, weaker statement is as follows.
One can easily check that it is still enough
for all the applications in <cit.>.


Consider α,β∈Φ_P satisfying α+β∈Φ_P and mα≠ -kβ for any m,k≥ 1.
Denote by Φ^0 an irreducible component of Φ such that α,β∈π(Φ^0).

(1) In each of the following cases,
(N_α,β,1,1) generates V_α+β as an abelian group:

  (a) structure constants of Φ^0 are invertible in R (for example,
if Φ^0 is simply laced);

  (b) α≠β and α-β∉Φ_P;

  (c) Φ^0 is of type B_l, C_l, or F_4,
and π^-1(α+β) consists of short roots;

  (d) Φ^0 is of type B_l, C_l, or F_4, and there exist long roots
α∈π^-1(α), β∈π^-1(β) such that α+β is a root.

(2) If α-β∈Φ_P and Φ^0≠ G_2, then 
(N_α,β,1,1),  ( N_α-β,2β,1,1), and 
(N_α-β,β,1,2(-,v)) for all v∈ V_β together
generate V_α+β as an abelian group. Here we assume
(N_α-β,2β,1,1)=0 if 2β∉Φ_P.





§ REDUCTION TO EXTRACTION OF UNIPOTENTS


In this section we establish two important reduction statements.
Let P be a proper parabolic R-subgroup of G,
Φ_P=Φ_J,Γ the system of relative roots for P, and  _α, α∈Φ_P,
the relative root subschemes of G with respect to P, which exist by Lemmas <ref> and <ref>.
Recall that for any ideal  of R we set U_P()=U_P(R)∩ G(R,) and
denote by E_P(R,) the normal closure of ⟨ U_P^+(),U_P^-()⟩ in E_P(R).

First we prove the standard commutator formulas, which are used for the reduction and complements the
normal structure theorem. The condition for inclusions in the following statement is the same as for
normality of the elementary subgroup in Theorem <ref>. It is easy to see that this condition
for a ring R is inherited by any R-algebra.



Let  be an ideal of a ring R.
Assume that for every maximal ideal  of R every normal semisimple
subgroup of G_R_ contains (𝔾_m,R_)^2. Then E_P(R,) does not depend on
a strictly proper parabolic subgroup P and is normal in G(R) and

    [G(R,),E(R)]≤ E_P(R,).




Let P and Q be two strictly proper parabolic subgroups of G.
To prove the first assertion in suffices to show that X_α(rv)∈ E_Q(R,) for any
α∈Φ_P, r∈, and v∈ V_α. Consider an element
X_α(tv)∈ E_P(R[t],tR[t]), where t is an independent variable.
Since tR[t] is a splitting ideal, by the splitting priciple for isotropic groups <cit.>
we have E_P(R[t],tR[t])=G(R[t],tR[t])∩ E_P(R[t]). The latter does not depend on the choice of a
parabolic subgroup P by the main theorem of <cit.>. Thus
X_α(tv)∈ E_Q(R[t],tR[t]). Specializing
t to r we get the required inclusion. The normality of E(R,) follows from this inclusion
applied to conjugate parabolic subgroups, and the normality of E(R).

The proof of the last statement is borrowed from <cit.>.
Let A=R[G] be the affine algebra of the scheme G, I its fundumental ideal, g the generic
element of G, and r∈ R. We identify elements of R and G(R) with their the canonical images in
A and G(A) respectively. Consider the commutator
h=[X_α(r),g]∈ G(A). Since the elementary subgroup is normal, we have
h∈ E(A). On the other hand, h vanishes modulo I and
I is a splitting ideal, i. e. A=R⊕ I as additive groups.
The splitting principle implies that
h∈ E(A)∩ G(A,I)=E(A,I).

Now, let b∈ G(R,).
We have

    b(h)=[X_α(r),b]∈ b(E(A,I))=E(b(A),b(I))≤ E(R,)

by Lemma <ref>. Thus, [G(R,),E(R)]≤ E(R,).


In the setting of Lemma <ref> we denote E_P(R,) by E(R,).


We say that the normal structure of the group G(R) is standard if for each subgroup H≤ G(R)
normalized by E(R), there exists a unique ideal  of R such that

    E(R,)≤ H≤ C(R,).



In order to secure the existence of relative roots and relative root subschemes as in Lemmas <ref> and <ref>,
assume from now on that
G is a reductive group scheme over a connected commutative ring R.
We are going to deduce the existence of the standard normal structure from the fact that any H
as above contains an elementary root unipotent.
Our starting point is the following theorem established by the first author.


Assume that the structure constants of the absolute root system Φ of G are invertible in R,
and for any maximal ideal  of R every semisimple normal subgroup of G_R_ contains
(𝔾_m_,R_)^2.
Let P be a strictly proper parabolic R-subgroup of G.
Then for any normal subgroup N≤ E(R) there exists an ideal
=(N) in R such that N∩_α(R)=X_α( V_α)
for any α∈Φ_P.


The following equalities are called the standard commutator formulas.


In the setting of Theorem <ref> we have

    [C(R,),E(R)]=[G(R,),E(R)]=[G(R),E(R,)]=[E(R),E(R,)]=E(R,).




First we prove the inclusion E(R,)≤[E(R),E(R,)].
By Theorem <ref> it is enough to show that
X_α̃(V_α̃⊗_R )⊆ [E(R),E(R,)], where α̃∈Φ_P^+ is a maximal root.
Let β∈Φ_P^+ be a simple relative root such that α̃-β∈Φ_P. Then by Lemma <ref>
and the generalized Chevalley commutator formula
X_α̃(V_α̃⊗_R )
is generated by all commutators [X_β(u),X_α̃-β(v)],
where v∈ V_α̃-β, u∈ V_β⊗_R.

In view of Lemma <ref> it remains to prove that [C(R,),E(R)]=E(R,).
The proof or this formula is an easy group theoretical excersise,
we reproduce it for the sake of completeness.
By the definition of C(R,) the left hand side of the formula is contained
in G(G,) and we have just proven that it contains E(R,).
Therefore, [C(R,),E(R)],E(R)]=E(R,). Now the result follows from
Lemma 1 of <cit.>, which asserts that [[H,D],D]=[H,D]
if D is a perfect subgroup and  [[H,D],D] is normal in H.


Using Theorem <ref>, we prove the following statement. The idea of this reduction goes back
to <cit.>.









In the setting of Theorem <ref>,
suppose further that every irreducible component of Φ_P contains more than 2 distinct
roots. If for any ideal  of R any
noncentral subgroup H≤ G(R/) normalized by E(R/)
contains a nontrivial root unipotent element X_α(u), α∈Φ_P, u∈ V_α⊗_R R/{0},
then the normal structure of the group G(R) is standard.


To prove Proposition <ref> we need the following technical statement.


Let α,β∈Φ_P be two
relative roots such that α+β∈Φ_P and nα≠ -mβ for all n,m≥ 1.
Assume, moreover, that α-β∉Φ_P or the structure constants of the absolute root system
Φ of G are invertible in R. Take u∈ V_β{0}. Any generating system
e_1,…,e_n of the R-module V_α contains an element e_i such that N_α,β,1,1(e_i,u)≠ 0.




Consider an affine fpqc-covering ∐Spec R_τ→Spec R that splits G. There is a member
R_τ=A of this covering such that the image of X_β(u) under G(R)→ G(A) is non-trivial.
Write

    X_β(u)=∏_π(ν)=βx_ν(a_ν)·∏_i≥ 2∏_π(ν)=iβx_ν(c_ν),

where π:Φ→Φ_P∪{0} is the canonical projection of the absolute root system of G onto the relative one,
x_ν are root subgroups of the split group G_A,
and a_ν∈ R. Since X_β(u)≠ 0, the definition of X_β implies that there exists
a_ν≠ 0.
By <cit.> there exists
a root μ∈π^-1(α) such that μ+ν∈Φ. Let v∈ V_α⊗_R A
be such that
X_α(v)=x_μ(1)∏_i≥ 2∏_π(η)=iαx_η(d_η),
for some d_η∈ R.
Then the (usual) Chevalley commutator formula implies that [X_α(v),X_β(u)] contains in its decomposition a
factor x_μ+ν( a_ν), where  is a structure constant
of Φ.

If α-β∉Φ_P, then =± 1, otherwise  is invertible by assumptions.
Hence N_α,β,1,1(v,u)≠ 0. Since N_α,β,1,1(v,u) is linear in the first argument,
this implies the result.



Let N be a subgroup of G(R) normalized by E(R).
If N≤(G(R)), there is nothing to prove. Otherwise by our assumption and
Theorem <ref> there is an ideal ≠ 0 of R such that N∩_α(R)=X_α( V_α)
for all α∈Φ_P. Then, clearly, E_P(R,)≤ N. If N is not contained in C(R,), then
ρ_(N) is a non-central subgroup of G(R/) normalized by E(R/), and hence by the same token we have
N≥ E_P(R/,') for some ideal '≠ 0 of R/. Let α̃∈Φ_P^+ be a maximal root,
and let β∈Φ_P^+ be a simple relative root such that α̃-β∈Φ_P. Pick
u∈ V_α̃-β{0} such that ρ_(u)∈ V_α̃-β⊗_R '{0}
and
X_α̃-β(ρ_(u))∈ρ_q(N). Then there is h∈ G(R,) such that
X_α̃-β(u)h∈ N.
By Lemma <ref> for any generating system
e_1,…,e_n of V_β
there is 1≤ i≤ n such that N_α̃-β,β, 1,1(ρ_(u),ρ_(e_i))≠ 0,
and hence N_α̃-β,β, 1,1(u,e_i)∉ V_α̃.
By Lemma <ref> one has

    [X_α̃-β(u)h,X_β(e)]=[h,X_β(e)]^X_α̃-β(-u)·
    X_α̃(N_α̃-β,β, 1,1(u,e_i))∈
    E_P(R,)· X_α̃(N_α̃-β,β, 1,1(u,e_i)).

Hence X_α̃(N_α̃-β,β, 1,1(u,e_i))∈ N. However, this contradicts the choice of .


Thus, we have reduced the proof of the normal structure theorem to extraction of unipotents from E(R)-normalized subgroups.
In the following section we show that one can always extract a nontrivial root unipotent from an element that belongs to
a proper parabolic subgroup.




§ INSIDE A PARABOLIC SUBGROUP


As before, assume that G is a reductive group scheme over a connected ring R.
We start with definitions of certain normal subgroups of the unipotent radical of a parabolic subgroup P of G.
Choose a set Π_P of simple relative roots and fix a total order on the relative root lattice
Φ_P in the following way. First, order Π_P arbitraraly. Then set α≥β
if the coordinate columns α_Π_P≥β_Π_P with respect to
the lexicographical order.
Note that for this order we have


    α>γ-α<-γα+β>γ+β


for all α,β,γ∈Φ_P.

For α∈Φ_P put

    Φ_P^>α={γ∈Φ_P|γ>α} and Φ_P^<α={γ∈Φ_P|γ<α}.

If α>0, then the set Φ_P^>α is a closed set of roots and we denote by U_P^>α
the subgroup scheme corresponding to Φ_P^>α (see Lemma <ref>).
Similarly, if α<0, then U_P^<α is the subgroup scheme corresponding to the closed set of roots
Φ_P^<α.
If α is the largest positive root of a given height m, then we denote U_P^>α by U_P^(m+1).
In other words, U_P^(m+1)(R) is the subgroup of U_P(R) generated by all root subgroups _γ(R) with
htγ>m.


Let α,β,α+β∈Φ_P^+, then
the subgroup U_P^>α(R) is normal in P(R) and, moreover,

    [U_P^>α(R),_β(R)]≤ U_P^>α+β(R).


Similarly, for α,β,α+β∈Φ_P^- the subgroup U_P^<α(R) is normal in P^-(R) and

    [U_P^<α(R),_β(R)]≤ U_P^<α+β(R).




We prove the first formula by going down induction on α.
If α is the maximal root, then U_P^>α={1}. Otherwise,
let γ be the smallest root greater than α.
By Lemma <ref>(iv) U_P^>α(R)=_γ(R)U_P^>γ(R).
Then

    [U_P^>α(R),_β(R)]=
    _γ(R)[U_P^>γ(R),_β(R)]·[_γ(R),_β(R)].

If β>0, then by induction hypothesis  the first factor is contained in U_P^>γ+β(R)
whereas the second one lies in _γ+β(R)U_P^>γ+β(R) by generalized Chevalley commutator formula (<ref>). Hence U_P^>α(R) is normal in U_P(R). With this in mind
for an arbitrary β the above arguments show that both factors lie in U_P^>α+β(R) as required.

Since P is generated by L_P and U_P and L_P normalizes each relative root subgroup, we conclude that
U_P^>α(R) is normal in P(R).
The proof of the second formula is essentially the same.


For the rest of the article we assume that each relative root system is endowed with
a total ordering, satisfying conditions (<ref>).


Suppose that the absolute root system of G is irreducible and its structure constants
are invertible in R. Let P be a parabolic R-subgroup of G with a Levi subgroup L_P,
such that rank(Φ_P)≥ 2.
For any h∈ U_P(R){1}, the subgroup
[h,E(R)] contains a root unipotent element X_α(v), α∈Φ_P^+,
v∈ V_α{0}.



By Lemma <ref>(iv)  we can write h as a product

    h=∏_α∈Φ_P^+X_α(v_α),   v_α∈ V_α,

where the order of factors coincides with the ascending order of roots. Let γ=γ(h)∈Φ_P^+ be
the minimal root such that v_γ≠ 0.
If γ is the maximal root of Φ_P^+, then h=X_γ(v_γ).
Since rank(Φ_P)≥ 2, there is a simple root β∈Φ_P^+ such that γ-β is a root
and all roots of shape iγ-jβ are positive.
By Lemma <ref> there is v∈ V_β such that
N_γ,-β,1,1(v_γ,v)≠ 0. Then h'=[h,X_-β(v)]∈ U_P(R){1}
and γ(h') is not maximal. Since [h',E(R)]≤[h,E(R)] (see remark after formula (<ref>)),
we may assume from the beginning that γ(h) is not maximal.

Under this assumption write h=X_γ(v_γ)g for some g∈ U_P^>γ(R).
There is a root β∈Φ_P^+ such that γ+β∈Φ_P^+.
By Lemma <ref> there is v∈ V_β such that N_γ,β,1,1(v_γ,v)≠ 0.
The commutator

    c=[h,X_β(v)]=X_γ(v_γ)[g,X_β(v)]· [X_γ(v_γ),X_β(v)].

By Lemma <ref> the first factor belongs to U_P^>γ+β(R) and by the generalized Chevalley commutator formula
the second one is equivalent to X_γ+β(N_γ,β,1,1(v_γ,v))
modulo U_P^>γ+β(R). Thus, c∈ U_P(R)∩[h,E(R)] and γ(c)=γ(h)+β>γ(h).
Therefore, we may proceed by
induction to arrive at the case, where γ(c) is the maximal root, i. e. c=X_γ(c)(u)∈[h,E(R)]
for some u∈ V_γ(c){0}.



Let P be a stricly proper parabolic subgroup of G with a Levi subgroup
L_P. Let z∈ L_P(R)
be such that [z,U_P(R)]⊆ U_P^(2)(R). Then z∈(G)(R).



Replace G by G^ad=G/(G). Then we need to show that z=1.
By Lemma <ref> for any α∈Φ_P there exists a set of degree i homogeneous polynomial maps
φ^i_z,α: V_α→ V_iα, i≥ 1, such that for
any commutative
R-algebra A and
any w∈ V_α⊗_R A one has

    zX_α(w)z^-1=∏_i≥ 1X_iα(φ^i_z,α(w)).

If α∈Φ_P^+ has height >1, then zX_α(w)z^-1∈ U_P^(2)(A) automatically. Thus, the inclusion
[z,U_P(R)]⊆ U_P^(2)(R) is equivalent to φ^1_z,α(v)=v for any α of height 1 and any
v∈ V_α.
Since φ^1_z,α is linear, φ^1_z,α(v)=v for any v∈ V_α
implies φ^1_z,α(w)=w for any w∈ V_α⊗_R A.

In other words, the conjugation action of L_P on U_P
induces a linear group scheme representation ρ:L_P→(V), where
V=⊕_htα=1 V_α, via the natural isomorphism U_P/U_P^(2)≅ W(V).
By our assumption, we have ρ(z)=∑_htα=1φ^1_z,α=id_V,
i.e. z∈ρ(R). Clearly, ρ is a closed subgroup scheme of  L_P.
We intend to show that it is trivial.

Since L_P is a group scheme locally of finite type over R, ρ is also locally of finite type. Then
by <cit.> in order to prove that ρ=1, it is enough to prove
that (ρ)_k(𝔭)=1 for every 𝔭∈Spec R. For the latter it suffices to
show that the scheme (ρ)_F is trivial, where F=k(𝔭) is the algebraic closure of
k(𝔭).

Thus, without loss of generality we may assume that G is a split reductive group over an algebraically closed field R=F,
and P is a standard parabolic subgroup of G with respect to a split maximal F-torus T≤ L_P.
We denote by Φ the root system of G over F, and P has type J, as in Lemma <ref>.
Every normal subgroup of L_P(F) is either central in L_P(F) or contains the elementary subgroup, corresponding to a whole irreducible component
of the root system of L_P(F) (which is non-trivial unless L_P=T), e.g. by <cit.>.
Clearly, ρ(F) does not contain the elementary subgroup of this form.

Indeed, since the sum of all simple roots of Φ is a root,
for any simple root β∉J there exists a root α∈π^-1(htα=1) such that
α+β∈Φ and
α-β∉Φ; then [x_β(1),x_α(1)]≠ 1. If j is a maximal positive integer such
that α+jβ∈Φ, then
α+jβ-(-β)∉Φ, and hence [x_-β(1),x_α+jβ(1)]≠ 1.
Then ρ(F)≤(L_P(F)). One has (L_P(F))=(L_P)(F)≤ T(F) (e.g. by <cit.>),
hence ρ(F)≤ T(F).

Similarly, (ρ)≤(T). Indeed,
the Lie algebra (L_P), and hence its subalgebra (ρ), decomposes into a direct sum of (T)
and 1-dimensional
root eigenspaces of T. Since ρ is a normal subgroup of L_P,
its Lie algebra is invariant under the adjoint action of L_P(F) on (L_P), and in particular
under the action of the Weyl group of L_P(F).
Hence (ρ) either contains no root subspaces at all, or contains all root subspaces corresponding to one
irreducible component of the root system of L_P. One proves similarly to the above that it cannot contain such
a set of root subspaces.

Let A be any F-algebra, then
any x∈ T(A) acts on root elements x_δ(a), δ∈Φ, a∈ A, by means
of a character χ of the root lattice of G (recall that G is adjoint) with values in A^×.
Since any α∈Φ_P^+ is a sum of simple
roots β_i∈Φ_P^+, by <cit.> any δ∈π^-1(α) is a sum of roots
δ_i∈π^-1(β_i). If x∈ρ(A), we have χ(δ_i)=1 for all i, hence χ(±δ)=1
for any δ∈π^-1(Φ_P^+), and thus x=1. This implies that both ρ(F)=1 and (ρ)=0.
In particular, (ρ)=((ρ)), which implies that ρ is smooth. Since
ρ(F)=1, we have ρ=1, as required.


For a relative root γ∈Φ_P denote by U_(γ) the unipotent subgroup generated by
_iγ for all i∈ such that iγ∈Φ_P. In other words, if
we set (γ)=γ∩Φ_P, then U_(γ) is the group scheme introduced in
Lemma <ref>.



Let P be a not necessarily strictly proper parabolic subgroup of G. Let β∈Φ_P^+ be a simple root.
If g∈ L_P(R) satisfies [g,_β(R)]≤ U_P^>β(R), then [g,U_(β)(R)]=1 and [g,U_(-β)(R)]=1.


Let H=U_β∩Φ_P, the reductive subgroup scheme of G that exists by Lemma <ref>. By the same Lemma,
it has two opposite
proper parabolic subgroups Q=L_PU_(β) and Q^-=L_QU_(-β) with the common Levi subgroup L_Q=L_P.
The assumption of the Lemma then implies [g,U_Q(R)]≤ U_Q^(2)(R).

Note that it is enough to prove the claim for H^ad instead of G. Indeed, if it holds for H^ad, then
[g,U_(±β)]≤ C(H)(R). However, [g,U_(±β)(R)]≤ U_(±β)(R) and
C(H)(R)∩ U_(±β)(R)=1, hence [g,U_(±β)(R)]=1.

Assume that G=H^ad. Since R is connected, by <cit.> G is a finite direct product of
adjoint semisimple
reductive R-groups G_i, 1≤ i≤ n, such that every G_i itself does not have any non-trivial normal
semi-simple subgroups. Consider the action of the torus S≤ C(L_P) constructed in Lemma <ref> on G_i.
If this action is non-trivial, then the projection of g into G_i(R) is trivial by Lemma <ref>, since
the projection of Q is a proper parabolic subgroup of G_i. Hence g belongs to the product of groups G_i(R)
such that S centralizes G_i, and, in particular, G_i≤ L_P. Then, since G_i is normal in G,

    [G_i(R),U_(β)(R)]≤ G_i(R)∩ U_(±β)(R)≤ L_P(R)∩  U_(±β)(R) =1,

and hence g centralizes U_(±β)(R).







Suppose that the absolute root system of G is irreducible and its structure constants
are invertible in R. Let P be a parabolic R-subgroup of G with a Levi subgroup L_P,
such that rank(Φ_P)≥ 2.
Let H be a subgroup of G(R) normalized by the elementary subgroup E(R).
Suppose that H∩ P(R) is not contained in (G(R)).
Then H contains a root unipotent element X_α(v), α∈Φ_P,
v∈ V_α{0}.



This is proved similarly to the corresponding statement <cit.> for Chevalley groups.
Let g∈ H∩ P(R) be a non-central element.
Write g=zu, where z∈ L_P(R), u∈ U_P(R)=U_Φ_P^+(R).
For any α∈Φ_P^+ and v∈ V_α, one has

    [g,X_α(v)]=z[u,X_α(v)]·[z,X_α(v)].

Clearly, z[u,X_α(v)]∈ [U_P,U_P](R).
If [z,X_α(v)]∉[U_P,U_P](R), then 1≠ [g,X_α(v)]∈ U_P(R)∩ H. Then H
contains a root unipotent element by Lemma <ref>.

Assume that [z,X_α(v)]∈ [U_P,U_P](R)
for any α∈Φ_P^+ and v∈ V_α. Then [z,U_P(R)]≤ U_P^(2)(R),
and hence z∈(G)(R) by Lemma <ref>.
Then [g,E_P(R)]=[u,E_P(R)]. Then
[u,E_P(R)]≤ H contains a root unipotent element by Lemma <ref>.





§ EXTRACTION FROM THE MAIN GAUSS CELL


As before, G is a reductive group scheme over R
and P is a proper parabolic R-subgroup of G.
Consider the R-subscheme of G

    Ω_P=U_PL_PU_P^-.

This is an open subscheme isomorphic as an R-scheme
to the direct product U_P× L_P× U_P^-, see <cit.>.
We call Ω_P the main Gauss cell associated with P.

In this section we show how to extract a unipotent from Ω_P(R).
The following easy lemma will be used several times.


Let G be an affine scheme over a commutative ring R,  Ω an open
subscheme of G, and g∈ G(R). Then g∈Ω(R) if and only if for all maximal ideals
 of R one has ρ_(g)∈Ω(R/).



By definition of an open set Ω=D() for some ideal  of the affine algebra
A=R[G] of the scheme G. Then

    g∈Ω g()R=Rρ_(g()){0} for all maximal ideals  of R.

But the latter is equivalent to the condition ρ_(g)∈Ω(R/)
for all maximal ideals  of R.


From now and until the end of this section, assume that R is connected.


Let R be a field and P a parabolic subgroup of G over R.
If g∈ G(R) commutes with U_P(R), then g∈ P(R).



Let Q≤ P be a minimal parabolic subgroup of G over R, let L_Q be its Levi subgroup contained in L_P,
and let Q^- be the opposite minimal parabolic subgroup contained in P^-. Let S≤ L_Q be the maximal split
subtorus of (L_Q). Reduced Bruhat decomposition implies
that g=uwv, where u∈ U_Q(R), w∈ N_G(S)(R),
v∈ (U_Q)_w(R)={x∈ U_Q(R)| wxw^-1∈ U_Q^-(R)}, and u,v, and the class of w in the Weyl group of Φ_Q
are unique.
We have w∈ L_P(R) if and only if
w is a product of elementary reflections w_α_i for some simple roots α_i∈Φ_Q
such that X_α_i≤ L_P <cit.>.

Assume that w∉ L_P(R). Then there is a simple root α∈Φ_Q outside the root system of L_P
such that w(α)<0.



Then x=X_α(ξ), belongs to U_P(R) for all ξ∈ V_α. Since
[g,x]=1, we have x(uwv)=(uwv)x.
The rightmost factor in the reduced Bruhat decomposition of x(uwv)=(xu)wv equals v.

On the other hand, w(vx)w^-1∈ U_Q^-(R), i. e. vx∈(U_Q)_w(R), which means that the rightmost factor of the
reduced Bruhat decomposition of (uwv)x=uw(vx) equals vx. But the latter is in contrary to the uniqness of the decomposition.
Thus, w∈ L_P and, since u,v∈ U_Q≤ Q≤ P, we get the result.











Suppose that the structure constants of the absolute root
system of G are invertible in R. Let Q be a parabolic R-subgroup of G,
such that Φ_Q is irreducible and rank(Φ_Q)≥ 2.
Let β∈Φ_Q be a simple relative root.
If x∈ U_Q^±(R) commutes
with _β(R), then x=∏ X_α(u_α), where α ranges over the set of
roots of Φ_Q^± such that α+β∉Φ_Q∪{0}.



First, consider the case where x∈ U_Q(R).
Write x=∏_α∈Φ_Q^+ X_α(u_α),
where u_α∈ V_α and the order of factors coincides with the
descending order of roots. Let γ be the smallest root such that γ+β∈Φ_Q and u_γ0.
Then x=x'X_γ(u_γ)x”, where x'∈ U_Q^>γ(R) and x” commutes with _β(R).
By Lemma <ref> there is v∈ V_β such that
N_β,γ,1,1(v,u_γ)=w≠ 0.
Now, by formula (<ref>), Lemma <ref>, and generalized Chevalley commutator formula (<ref>)
we have

    [X_β(v),x]=[X_β(v),x']·x'[X_β(v),X_γ(u_γ)]≡ X_β+γ(w) U_Q^>β+γ(R).

Hence the commutator is not equal to 1. The contradiction shows
that α+β∉Φ_Q for all α∈Φ^+_Q such that u_α0.

Second, assume that x=∏ X_α(u_α), where α ranges over the set
Φ_Q^-·(-β), u_α∈ V_α, and the order of factors
coincides with the ascending order of roots.
In other words, in this case x lies in the unipotent radical of the smallest parabolic subgroup P
containing Q^- and _β.
Let γ be the largest root such that γ+β∈Φ and u_γ0.
Since β is a simple root and γ<0, the root γ+β is negative.
Write x=x'X_γ(u_γ)x”, where x'∈ U_Q^<γ and x” commutes with _β(R).
Similarly to the first pragraph of the proof one shows that the commutator
[x,X_β(v)]≢1 U_Q^<γ+β, which implies the result.

Finally, let x∈ U_Q^-(R) be an arbitrary element. Write x=x_1x_2, where x_1∈ U_Q^-(R)∩ L_P
(i. e. x_1 is a product of root elements corresponding to the roots in (-β))
and x_2∈ U_P.
For any v∈ V_β one has

    1=[X_β(v),x_1x_2]=[X_β(v),x_1]·x_1[X_β(v),x_2],

where [X_β(v),x_1]∈ L_P(R) and x_1[X_β(v),x_2]∈ U_P(R), hence

    1=[X_β(v),x_1]=[X_β(v),x_2].

By the previous case we conclude that x_2 is a product of root elements corresponding to α∈Φ_Q^-
such that α+β∉Φ_Q. By definition of x_2 we also have α+β≠ 0.
It remains to show that x_1=1.

Assume that x_1≠ 1. We can write x_1=∏_i=1^k X_-iβ(u_-iβ), u_-iβ∈ V_-iβ.
Since Φ_Q is irreducible and its rank ≥ 2, there exists
a simple relative root α∈Φ_Q^+ such that α+β∈Φ_Q. Then α+iβ∈Φ_Q
for any i≥ 1 such that iβ∈Φ_Q by Lemma <ref>.
Let j≥ 1 be the smallest natural number such that u_-jβ≠ 0. Since -jβ-α∈Φ_Q,
by Lemma <ref> there is
u∈ V_-α such that N_-jβ,-α,1,1(u_-jβ,u)=w≠ 0.
Set y=[x_1,X_-α(w)]. Similarly to the first paragraph of the proof
y equals X_-jβ-α(w) modulo U_Q^<-jβ-α.

On the other hand, β-α∉Φ_Q as α and β are simple roots.
Hence [X_β(v),X_-α(u)]=1 for any v∈ V_β. Therefore,

    [X_β(v),y]=[X_β(v),[x_1,X_-α(w)]]=1

for any v∈ V_β. By the previous case this implies that y cannot contain
X_-jβ-α(u) with non-zero u in its reduced decomposition, a contradiction. This shows that x_1=1.




Under the hypothesis of Lemma <ref>, let m≥ 1 be the maximal positive integer such that
mβ∈Φ_Q. If x∈ U_(β)(R)L_Q(R)U_(-β)(R) commutes
with _β(R), then x∈_mβ(R)L_Q(R).



Assume that x=ahb for some a∈ U_(β)(R), h∈ L_Q(R), and b∈ U_(-β)(R){1}.
Put b'=hb^-1h^-1 and write it in the form b'=X_-jβ(u)b”, where u∈ V_-jβ{0} and
b”∈ U_Q^<-jβ.
Let α≠β∈Φ_Q be a simple relative root adjacent to β.
Since iβ-α∉Φ_Q (coefficients at simple roots have different signs),
the generalized Chevalley commutator formula implies that
[_-α(R),U_(β)(R)]=1. Hence,  _-α(R) commutes with a.
Since _β(R) commutes also with x, we have [[_-α(R),x^-1],_β(R)]=1.
On the other hand, by Lemma <ref>
-jβ-α∈Φ_Q. Therefore, by Lemma <ref> there is v∈ V_-α such that
N_-α,-jβ,1,1(u,v)≠ 0. Then,


    c=[X_-α(v),x^-1]=[X_-α(v),h^-1X_-jβ(u)b'a^-1]=
    
    [X_-α(v),h^-1]·
    [X_-α(v),X_-jβ(u)]^h·h^-1X_-jβ(u)[X_-α(v),b']


By Lemma <ref> the first factor belongs to U_(-α), whereas the second one is equal to
X_-α-jβ(w) modulo U_Q^<-α-jβ for some w∈ V_-α-jβ{0}.
By Lemma <ref> the last factor lies in U_Q^<-α-jβ.
Thus, c contains X_-α-jβ(w) in its reduced decomposition and by Lemma <ref>
cannot commute with X_β(R) as (-α-jβ)+β∈Φ_Q.
The contradiction shows that b=1, and x=ah, where a∈ U_(β)(R) and h∈ L_Q(R).
Then for every v∈ V_β we have

    1=[X_β(v),x^-1]=[X_β(v),h^-1]·[X_β(v),a^-1]^h.

By the generalized Chevalley commutator formula we have
[X_β(v),a^-1]^h∈ U_{kβ| k≥ 2}(R), and hence
[_β(R),h^-1]≤ U_{kβ| k≥ 2}(R). By Corollary <ref> this implies
that [h,U_(β)(R)]=1. Then [a,_β(R)]=1.
By Lemma <ref> this implies that a∈_mβ(R).


The following easy fact will be used twice in the proof of Lemma <ref>.


Let P be a parabolic subgroup of G
and g=ahu∈Ω_P(R), where a∈ U_P(R), h∈ L_P(R), and u∈ U_P^-(R).
If an element x∈ L_P(R) commutes with g, then it commutes with a, h, and u.



By the condition we have 1=[x,ahu]^a=[a^-1,x]·[x,hu].
Hence [x,a^-1]=[x,hu]∈ U_P(R)∩ P^-(R)={1}. Therefore,
x commutes with a and [x,hu]=[x,h]·h[x,u]=1. Now, [h,x]^h=[x,u]∈ L_P(R)∩ U_P^-(R)={1}.
Thus, x commutes with u and h as well.



Suppose that the absolute root system of G is irreducible and its structure constants
are invertible in R. Let Q be a parabolic R-subgroup of G with a Levi subgroup L_Q
such that rank(Φ_Q)≥ 2.
For a proper parabolic subgroup P≥ Q choose L_P≥ L_Q. Then, given g∈Ω_P(R)(G(R))
there exists a proper parabolic subgroup M≥ L_Q of G such that
[g^E(R)∩ M(R),E(R)]≠ 1.















By Theorem <ref> (G(R))=(G)(R) is contained in every parabolic subgroup, hence
by Lemma <ref> it is enough to prove the claim under the assumption that G is adjoint.
Let β∈Φ^Q be a simple relative root. Denote by Q_(β) the smallest parabolic subgroup
of G containing Q and _-β. Note that since rank(Φ_Q)≥ 2, the parabolic subgroup Q_(β)
is proper. In case P=Q we substitute P by Q_(β) using the fact that
Ω_Q⊆Ω_Q_(β). Thus, we may assume that L_P contains _β for some simple
relative root β∈Φ_Q.

Write g=ahu, where a∈ U_P(R)≤ U_Q(R) and h∈ L_P(R), and u∈ U_P^-(R).

Denote by lev_P(g) the largest vector α∈Φ_Q^+∪{0} such that a∈ U_Q^>α(R).
We proceed by going down induction on lev_P(g). If lev_P(g) is the maximal relative root,
then a=1 and g already lies in the parabolic subgroup P^-. Otherwise,
consider the element

    g'=[X_β(v),g]^a=[a^-1,X_β(v)]· [X_β(v),b]∈ g^E(R).

Here [X_β(v),b]∈ P^-(R), while [a^-1,X_β(v)]∈ U_Q^>α+β(R).
Thus, g'∈Ω_P(R) and lev_P(g')>lev_P(g). Hence, if g' does not belong to
the center at least for one simple root β∈Φ_Q such that _β≤ L_P and
v∈ V_β, we are done by the induction hypothesis.

Now, assume that g' lies in the center of G(R) for all β as above and all v∈ V_β. Since G is adjoint,
we have g'=1, i. e. _β(R) commutes with g. By Lemma <ref>
_β(R) commutes with a, h, and u as well.
Denote by Q' the parabloic subgroup Q∩ L_P of the reductive group L_P and
let Δ_Q' be the root system of L_P relative to Q'.
Note that h∈ L_P commutes with _β(R) for all simple roots β∈Δ_Q'.
Therefore, for any maximal ideal  of R the image of h in L_P(R/)
commutes with _β(R/).
Since the structure constants of the absolute root system of G are invertible in R,
the same is true for the absolute root system of L_P.
By Lemma <ref> the groups _β(R/) span U_Q'(R/) as β ranges
over all simple roots of Δ_Q'. Thus, the image of h in L_P(R/) commutes with
U_Q'(R/), and, hence, belongs to Q'(R/)⊆Ω_Q'(R/) by Lemma <ref>.

Since Ω_Q' is an open subscheme of L_P,
by Lemma <ref> we have h∈Ω_Q'(R).
Then

    g=ahu∈ U_P(R)·Ω_Q'(R)· U_P^-(R)⊆Ω_Q(R).


Let P' be the minimal parabolic subgroups of G, containing Q and _-β.
Since g∈Ω_Q(R)⊆Ω_P'(R), we can write g=a'h'u' for some
a'∈ U_P', h'∈ L_P', and u'∈ U^-_P'. Recall that g commutes with
_β(R). Then, Lemma <ref> asserts that a', h', and u' commute with
_β(R) as well.
By Lemmas <ref>
and <ref> this implies that g is a product of root elements of the form X_α(u_α) where
all α satisfy α+β∉Φ_Q, and of an element of L_Q(R). Then by Lemma <ref>
there is a proper parabolic set of relative roots Σ⊂Φ_Q such that g∈ U_Σ(R) in the notation
of Lemma <ref>. Then we take M=U_Σ.


The following statements follow immediately from Lemmas <ref> and <ref>.


Under assumptions of Lemma <ref> the subgroup g^E(R) contains a nontrivial root unipotent element.



Suppose that G is semisimple, R is connected, the absolute root system Φ of G is irreducible,
its structure constants are invertible in R, and G contains (𝔾_m,R)^2.
If H∩ G(R,RadR)⊈(G(R)),
then H contains a nontrivial root unipotent element.



Since G contains (𝔾_m,R)^2, there is a parabolic subgroup Q of G such that
rank(Φ_Q)≥ 2. Take g∈ (H∩ G(R,RadR))∖(G(R)).
Then ρ_(g)∈Ω_Q(R/)
for any maximal ideal  of R. Then by Lemma <ref> g∈Ω_Q(R),
and the result follows from Corollary <ref>.






§ PROOF OF THE NORMAL STRUCTURE THEOREM


The next lemma is a key step to the proof of the normal structure theorem.


Let G be a reductive group scheme over a Noetherian connected ring K, and let Q be a parabolic K-subgroup of G such that
Φ_Q is irreducible and rank(Φ_Q)≥ 2.
For any parabolic subgroup Q≨ P
there exist elements c_ij∈ g^E(A)∩Ω_P(A), i=1,…,l, j=1,…,n,
satisfying the following property.
Let S be a subscheme defined by the formula

    S(R)={h∈ G(R)| h(c_ij)∈(G)(R) for all i=1,…,l and j=1,…,n}

for any K-algebra R.
Then the group of points S(F) does not contain E(F) for any K-algebra F
that is a field.



Since the subscheme Ω_P is open, hence it is covered by principal open subschemes
Sp_K A_s_i, i=1,…,l. Let α,β∈Φ_Q^- be two relative roots such that -α,-β are simple roots,
α+β∈Φ_Q, _α⊆ L_P, _β⊆ U_P^-.
Let {e_1,…,e_n} be a set of generators of the K-module V_α.
Choose i=1,…,l and j=1,…,n and put s=s_i and v=e_j.

Let g_s be the image of g in G(A_s). In other words, g_s=λ_s is the localization
homomorphism. It is a tautology that g_s factors through A_s, which means that
g_s∈Sp_KA_s(A_s)⊆Ω_P(A_s).
Decompose it into a product g_s=ab, where  a∈ U_P(A_s) and b∈ P^-(A_s).
Since A is Noetherian, by Lemma <ref> there exists m∈ such that the restriction of λ_s to
s^mA is injective.
By Lemma <ref>
there exists a positive integer k≥ m such that

    d_s=aX_α(s^kv)∈λ_s(U_P(A,s^mA))
     and 
    f_s=d_s^g_s=X_α(s^kv)^b∈λ_s(P^-(A,s^mA)).

Let d=λ_s^-1(d_s)∈ U_P(A,s^mA) and f=λ_s^-1(f_s)∈ P^-(A,s^mA)
(by definition of m these preimages are unique).
Put c_ij=c=[g^-1,d]∈ G(A,s^mA). Then

    λ_s(c)=[g_s^-1,d_s]=d_s^g_sd^-1=f_sd_s^-1∈λ_s(Ω_P(A)).

By definition of m we have c_ij=fd^-1∈Ω_P(A).

Let F be a field.
Put h=X_β(u)∈Ω_P(F)=⋃_i=1^lSp_KA_s_i(F)
for some u∈ V_β⊗_KF{0}. Choose i such that h∈Sp_KA_s_i and put
s=s_i. Then h factors through A_s, i. e. h= h∘λ_s for some
h:A_s→ F. Since h(g)=h, we have
h= h(g_s)= h(a) h(b)=e· X_β(u).
The uniqueness of representation of h as a product
of an element from U_P(F) by an element from P^-(F) implies that h(a)=e.
Thus, we get


    h(c_ij)=[h(g^-1),h(d)]=[X_β(u)^-1,X_α(s^ke_j)].


By Lemma <ref> there exists j such that N_α,β,1,1(e_j,u) 0.
By the generalized Chevalley commutator formula h(c_ij)∉(G)(F). Thus,
h∉ S(F), and hence S(F) does not contain E(F), as required.


Now we are ready to prove the main theorem of the present paper.


Let T≤ G be the subgroup isomorphic to (𝔾_m,K)^2.
By Lemma <ref> there are two parabolic subgroups Q=Q^+ and Q^- of G
with a common Levi subgroup L_Q=_G(T).

Assume first that K is Noetherian. Then K is a finite product of connected Noetherian rings, so we can assume without loss
of generality that K is Noetherian and connected.  The relative root system Φ_Q of Lemma <ref>
is irreducible and rank(Φ_Q)≥ 2. By Proposition <ref> we have E(K,)=[G(K,),E(K)].

Thus,
by Proposition <ref> it suffices to prove that if H is not inside (G(K)), then
it contains a nontrivial relative root unipotent element.

Let β∈Φ_Q be a simple root. Then by Lemma <ref>
P=P^+=U_Φ_Q^+∪β is a parabolic subgroup distinct from
Q, with a Levi subgroup L_P=U_β. Let
S be a subscheme of G satisfying conditions of Lemma <ref>.
If there exists h∈ H such that h(c_ij)∉(G(K)), then h(c_ij)∈ h^E(K)∩Ω_P(K)
is subject to Corollary <ref>, and hence H contains a nontrivial
relative root unipotent element.

Otherwise H is contained in the set of K-points of the subscheme S.
Consider a maximal ideal  of K. The image H of the subgroup H under the canonical
homomorphism G(K)→ G(K/) is contained in S(K/) and is normalized by
E(K/). Tits' simplicity theorem <cit.> then implies that either H contains E(K/), or H is contained in
(G(K/)). Since E(K/) is not contained in S(K/),
H is contained in (G(K/)).
It follows that the image of the subgroup
[H,E(K)] vanishes in G(K/), i. e. [H,E(K)]≤ G(K,).

Since  is an arbitrary maximal ideal, we have [H,E(K)]⊆ G(K,J), where J is the Jacobson
radical of K. On the other hand, by Lemma <ref> [H,E(K)] is not contained in the center of G(K).
Hence, by Corollary <ref> this subgroup contains a nontrivial relative root unipotent element.
This completes the proof of the Noetherian case.

Now let K be arbitrary. For any finite set of elements Λ⊆ H there is a finitely generated subring
K of K and a semisimple reductive group scheme G over K with a subgroup T≅ (𝔾_m, K)^2
such that G= G_K, T= T_K,
and Λ⊆ G( K). Clearly, by Lemma <ref> there is also a parabolic subgroup Q≤ G
such that Q= Q_K.

Since for any maximal ideal  of K there is a maximal ideal
 of K such that ⊆, we conclude that G( K/) also has irreducible root system whose structure
constants are invertible in K. Thus Λ^E( K)≤ G( K) is subject to the Noetherian case of the theorem.
Hence there exists an ideal (Λ)⊆ K such that

    E( K,(Λ))≤Λ^E( K)≤ C( K,(Λ)).

Clearly, (Λ) is uniquely determined by K and Λ.
Let  be the ideal of K generated by all subsets (Λ)⊆ K⊆ K. Then, clearly, H≤ C(K,).
W show that E_Q(K,)≤ H. In order to do that,
it is enough to check that
U_Q^±((a)) is contained in H for any finite K-linear combination a=∑ c_ia_i of elements a_i∈(Λ_i).
Let K be subring of K corresponding to the finite set
Λ=∪Λ_i, and let K' be the subring of K generated by K and all c_i. Then by the Noetherian
case of the theorem applied to G_K', we conclude that
U_ Q^±((a))=U_Q^±((a)) are contained in Λ^E(K')≤Λ^E(K)≤ H.

The same argument as above also shows that E_Q(K,)=[G(K,),E(K)] and E_P(K,)=E_Q(K,) for any other parabolic subgroup P of G, since this
equality holds for each finitely generated subring K such that P is defined over K and the ideal = K∩.





amsalpha


