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 $pneq 2$ the determinant takes two values on $O_q$, 1 and $-1$,
and therefore the subgroup $SO_q:=O_qcap SL_{2n}$ is of index 2 in $O_q$; it is known that $O_qcap SL_{2n}$ is connected.

In characteristic 2 the determinant takes only one value 1 on $O_q$ (because $-1=1$), and therefore $O_qcap SL_{2n}=O_q$.
Still there is a homomorphism $Dcolon O_qto mathbf{Z}/2mathbf{Z}$ 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

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