Let $k$ be a field, $n\in\mathbb{N}$ and $f:k^n\times k^n\to k$ a non-degenerate symmetric bilinear form. Let $$O_n(k,f):=\{ g\in GL_n(k) \mid \forall x,y\in k^n : f(x,y)=f(g.x,g.y) \}$$ and $$SO_n(k,f):=O_n^+(k,f):=O_n(k,f) \cap SL_n(k)$$ be the associated (special) orthogonal group, i.e. linear transformations which preserve the form (with determinant 1). According to a paper by María J. Wonenburger ("The automorphisms of the group of rotations and its projective group corresponding to quadratic forms of any index", Canad. J. Math. 15 (1963), 302–303), building upon the works of Dieudonné and Rickart, the following is true:

Theorem: Suppose $\operatorname{char} k\neq 2$ and $n\geq 5$. Then for every $\phi\in \operatorname{Aut}(SO_n(k,f))$ there exists a homomorphism $\chi:SO_n(k,f)\to \{\pm I_n\}$ and a semi-similitude $T$ of $f$ such that $$\phi(g)=\chi(g)\ TgT^{-1}.$$

Here, $I_n$ is the identity matrix, and a semi-similitude of $k^n$ is a permutation of $K^n$ for which there are $\sigma\in \operatorname{Aut}(k)$ and $\lambda \in k\setminus\{0\}$ such that for all $x,y\in k^n$ one has $$f(Tx,Ty) = \lambda\ f(x,y)^\sigma.$$

If we call automorphisms that can be written as in the theorem ``standard'', then the theorem shows that for $n\geq 5$, all automorphisms are standard automorphisms.

Question: What is known when $n=3$ and $n=4$? Are there any ``non-standard'' automorphisms? If yes, when do they exist and what do they look like?

In these cases, we are essential looking at rational points of an algebraic group of type $B_2$ resp. $D_2$.

Both (counter)examples as well as partial affirmative answers (e.g. with restrictions on the Witt index or on the the field) are of interest.