Exceptional isomorphisms of classical algebraic groups

Question

Let $k$ be an algebraically closed field of characteristic $p\geq 0$. An affine algebraic group $G$ is an affine variety over $k$ with a group structure such that multiplication and inversion are morphisms of varieties.

A simple algebraic groups is defined to have no non-trivial connected normal subgroup (it may have center, but finite!)

We have a classification of simple algebraic groups in terms of the Dynkin diagrams:

1. Classical: $A_n,B_n,C_n,D_n$;
2. Exceptional: ...

The following equalities of Dynkin diagrams are clear:

• $A_1=B_1=C_1$;
• $B_2=C_2$;
• $A_3=D_3$;
• $A_1\times A_1=D_2$

My question is: can we write explicit isomorphisms between small rank classical groups? (not necessary simple)

It is very easy to show $\mathrm{Sp}_2(k)=\mathrm{SL}_2(k)$ and $\mathrm{O}_2(k)\cong\mathrm{GL}_1(k).2$. (the previous symbols denote the special linear group, general linear group, symplectic group and orthogonal group).

For example, Exercise 1.8.19 in Geck "An intro to alg geom and alg groups" implies that $\mathrm{PSL}_2(k)\cong\mathrm{O}_3$, provided $p\neq2$, by constructing an explicit isomorphism (actually a surjective morphism whose kernes is $\{\pm1\}$).