In characteristic not $2$, the Theorem of Cartan-Dieudonné states:


1. [Grove, Theorem 6.6]: Let $q$ be a nondegenerate symmetric quadratic form of dimension $n$ in characteristic not $2$. Then every element in the orthogonal group $O(q)$ can be written as a product of at most $n$ reflections.


**In a nutshell:** I am looking for a similar statement but over a field in **characteristic $2$** and in particular in **odd dimension**. At the end I will make the statement that I would like to have more precise.


As I am new to quadratic forms, I got confused by different notions of non-degeneracy, so I want to fix the terminology; I take the one from [Grove]. To a quadratic form $q$ on $V$ one can associate a bilinear form $b$ on $V$ via $b(v,w)=q(v+w)-q(v)-q(w)$. In characteristic $2$, several quadratic forms can give the same bilinear form (e.g. $y^2+xz$ and $xz$).
Let $\mathrm{rad}\_b(V)=\{w\in V\mid b(v,w)=0 ~\forall v\in V\}$. If $\mathrm{rad}\_b(V)\neq\{0\}$, then $q$ is called


* *degenerate* if the characteristic is not $2$,
* *defective* if the characteristic is $2$.


**In characteristic $2$ and dimension odd, all quadratic forms are defective!**
In characteristic $2$, we say that $q$ is *regular* if $q(v)\neq0$ for all non-zero $v\in\mathrm{rad}(V)$. (In [Connors], regular is called *nondegenerate*).


**Example:** The two quadratic forms $q\_1=y^2+xz$ and $q\_2=xz$ on $V=k^3$, where $k$ is a field of characteristic $2$, give the same bilinear form and they both have $\mathrm{rad}(V)=\langle(0,1,0)\rangle$. So they are defective. While $q\_1$ is regular, $q\_2$ is not.


From now on, assume that $k$ is a field in characteristic $2$. Here transvections play the role of reflections. I have found the following statements that go in the right direction:


2. [Grove, Theorem 14.16]: Assume that $q$ is regular and **non-defective** (see the assumption on p. 129 which I believe is still taken in Theorem 14.16). Then $O(q)$ is generated by transvections, except for one example over the field $\mathbb{F}\_2$ where $V$ is $4$-dimensional.


The above theorem can not be applied in odd dimension. The next statement is exactly what I want, but it is stated only for dimension $3$:


3. [Connors, Proposition 3.2]: Let $q$ be a defective, anisotropic regular quadratic form in **dimension $3$**. Then every element of $O(q)$ is the product of at most $2$ transvections.


**I am looking for a reference of the following statement** (if true): Let $V$ be a vector space over any field in **characteristic $2$**, and let $q$ be a **regular**, **anisotropic** quadratic form on $V$ of **odd dimension** $n$ (i.e. $q$ **defective**). Then every element in the orthogonal group $O(q)$ can be written as a product of (at most $n$) transvections.


[Grove] : Larry C. Grove, "Classical groups and geometric algebra", 2002


[Connors] : Edward A. Connors, ''The Structure of $O'(V)/DO(V)$ in the Defective Case'', 1973



Generators of the orthogonal group of a quadratic form in odd dimension in characteristic 2
I suggest reading the result 6.2.17 in the book written by A.J.Hahn and O.T.O'Meara called "The Classical Groups and K-Theory". The result is a general version of the Cartan-Dieudonné-Scherk theorem (which implies the Cartan-Diuedonné theorem). The result is stated in the language of "Quadratic spaces over form rings". The case of the defective orthogonal groups is a particular case of the general "unitary groups" $U\_n(V)$. More precisely, in the book you find


6.2.17 Theorem. Suppose $U\_n(V)$ is not a symplectic group. If $J=\mathrm{id}\_R$, assume that $R$ is not $F\_2$ and if $J\neq \mathrm{id}\_R$, assume $R$ is not $F\_4$. Let $\sigma$ in $U\_n(V)$ be non-trivial and let $S$ be its residual space.


(i) If $\sigma$ is not totally isotropic, $\sigma$ is a product of $\mathrm{dim} S$ symmetries, but no fewer.


(ii) If $\sigma$ is totally isotropic, $\sigma$ is a product of $\mathrm{dim} S+2$ symmetries, but no fewer.


When $J=\mathrm{id}\_R$ you can get the defective orthogonal groups and so any $\sigma$ will be a product of "symmetries" (i.e. transvections in the case of the defective orthogonal group).


