Special unitary group of degree two

Definition
Suppose $$L$$ is a separable quadratic extension of a field $$K$$ and $$\sigma:L \to L$$ is the unique automorphism of $$L$$ that fixes $$K$$ pointwise. The special unitary group of degree two $$SU(2,L)$$ corresponding to this field extension is the special unitary group of degree two: it is the subgroup of the special linear group of degree two over $$L$$ comprising those matrices for which the entry-wise image under $$\sigma$$ coincides with the effects of the transpose-inverse map. Explicitly, this works out to be:

$$SU(2,L) := \left \{ \begin{pmatrix} a & b \\ -\sigma(b) & \sigma(a)\\\end{pmatrix} \mid a,b \in L, a\sigma(a) + b\sigma(b) = 1 \right \}$$

For the real and complex numbers
The most typical usage of the term special unitary group is in the context where $$K$$ is the field of real numbers, $$L$$ is the field of complex numbers, and the automorphism $$\sigma$$ is complex conjugation. In this case, the group is described explicitly as follows:

$$SU(2,\mathbb{C}) := \left \{ \begin{pmatrix} a & b \\ -\overline{b} & \overline{a} \\\end{pmatrix} \mid a,b \in \mathbb{C}, |a|^2 + |b|^2 = 1 \right \}$$

Here, $$\overline{a}$$ and $$\overline{b}$$ denote the complex conjugates of $$a$$ and $$b$$ respectively.

For a finite field
If $$K$$ is the (unique up to isomorphism) finite field of size a prime power $$q$$, there is a unique quadratic extension $$L$$ of $$K$$, and this extension is separable. The extension field is the finite field (unique up to isomorphism) of order $$q^2$$. The automorphism $$\sigma$$ is the map $$x \mapsto x^q$$. The special unitary group for this extension may be denoted $$SU(2,q)$$ (the more standard choice) or $$SU(2,q^2)$$ (a less standard choice). It is given explicitly as:

$$SU(2,q) := \left \{ \begin{pmatrix} a & b \\ -b^q & a^q \\\end{pmatrix} \mid a,b \in L, a^{q+1} + b^{q+1} = 1 \right \}$$

It turns out that this group is isomorphic to the special linear group of degree two $$SL(2,q)$$: special unitary group of degree two equals special linear group of degree two over a finite field.