I would like to find a reference for the following fact, which is sort of "well known":
Let $Q$ be a positive definite non-degenerate quadratic form in $n$ variables over an algebraically closed field $\mathbb{K}$. Let $SO_Q$ denote the special stabiliser group of the form $Q$:
$$ SO_Q = [ g \in GL_n(\mathbb{K}) : Q(gx) = Q(x) \ \text{for all} \ x \in \mathbb{K}^n \quad \text{and} \quad \det(g)=1 ]. $$
Let $SL_n$ be the special linear group:
$$ SL_n = [ g \in GL_n(\mathbb{K}) : \det(g)=1 ]. $$
I would like a reference for the following fact:
$$SO_Q \ \text{is maximal connected in} \ SL_n.$$
Here by maximal connected we mean that there does not exist a connected group $M \leq SL_n$ such that $M$ strictly contains $SO_Q$ and is strictly contained in $SL_n$.
Any reference would be of help. Thank you!
P.s.: I roughly know how to prove it, I would only be interested in knowing if someone has written down a proper proof of this.