I am looking for a reference for the following fact:
The orthogonal group $O\_{2n}$ over an algebraically closed field of characteristic 2
has exactly two connected components.


To be more precise, let $O\_q$ denote the orthogonal group of the quadratic form $q(x)=x\_1 x\_2 +x\_3 x\_4+\cdots +x\_{2n-1}x\_{2n}$
over an algebraically closed field $k$.
In characteristic $p\neq 2$ the determinant takes two values on $O\_q$, 1 and $-1$,
and therefore the subgroup $SO\_q:=O\_q\cap SL\_{2n}$ is of index 2 in $O\_q$; it is known that $O\_q\cap SL\_{2n}$ is connected.


In characteristic 2 the determinant takes only one value 1 on $O\_q$ (because $-1=1$), and therefore $O\_q\cap SL\_{2n}=O\_q$.
Still there is a homomorphism $D\colon O\_q\to \mathbf{Z}/2\mathbf{Z}$ given by a polynomial $D$ called the Dickson invariant,
see J.A.~Dieudonn\'e, Pseudo-discriminant and Dickson invariant, Pacific. J. Math. 5 (1955), 907--910.
This homomorphism $D$ indeed takes both values 0 and 1 on $O\_q$, and therefore its kernel ker $D$
is a closed subgroup of index 2 in $O\_q$. I would like to know that ker $D$ is connected.
In other words, I am looking for a reference to the assertion that the orthogonal group $O\_q$ has at most two connected components.
This is proved in Brian Conrad's handout "Properties of orthogonal groups" to his course Math 252 "Algebraic groups",
see <http://math.stanford.edu/~conrad/252Page/handouts/O(q).pdf> . Is there any other reference for this fact?


I will be grateful to any references, comments, etc.


Mikhail Borovoi



Connected components of the orthogonal group O(2n) in characteristic 2.
Presumably this is treated in detail in chapter 7 of the book 
The Classical Groups and K-Theory, by A.J.Hahn and O.T.O'Meara.


On page 424 it says in theorem 7.2.23 that the elementary subgroup has index two.
And elementary matrices are in the connected component of 1.


Wilberd


