Connected components of the orthogonal group O(2n) in characteristic 2. 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
 A: 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
A: When working with orthogonal groups and Spin groups, in arbitrary characteristic (over an arbitrary commutative base ring, in fact), I like the article "Clifford Algebras and Spinor Norms over a Commutative Ring", by Hyman Bass.
In particular, Bass describes a short exact (in the fppf topology over a base ring $R$) sequence:
$$1 \rightarrow \mu_2 \rightarrow Spin \rightarrow SO \rightarrow 1,$$
of group schemes, arising from a projective $R$-module equipped with a nonsingular $R$-valued quadratic form.  Here, the group $SO$ is defined as the kernel of the "degree" homomorphism from $O$ to the locally constant sheaf $Z / 2 Z$, which generalizes the determinant appropriately to arbitrary base rings (in particular, $det = (-1)^{deg}$, when $2$ is invertible).
So I think that connectedness follows from Bass's paper, given that the well-known group $Spin$ is connected.  
A: This is also described in section 23 of The Book of Involutions. 
A: An alternative presentation of Dickson's invariant is given in §4.1.2 of the article On the adjoint quotient of Chevalley groups over arbitrary base schemes, JIMJ 2010, by P.-E. Chaput and myself. There we show how the determinant $O(2n)\to\mathbb{Z}/2\mathbb{Z}$ in characteristic 0 (or say different from 2) extends to a morphism of group schemes over $\mathbb{Z}$, reducing to Dickson's invariant modulo the prime 2.
