Let $p>2$ be prime. By the classification of quadratic forms, there are $8$ pairwise non-equivalent isotropic orthogonal groups in $4$ variables. Is there a concrete classification of orthogonal groups of these groups? Could they be isomorphism? The analogous question for $3$ variables has been asked and answered in a previous question of mine ( [$p$-adic analogues of $SO(3)$](https://mathoverflow.net/questions/177847/p-adic-analogues-of-so3) ) but the method used there does not apply to forms with $4$ variables. 



$p$-adic orthogonal groups in four variables
There is a concrete description of the various forms in a paper by Harris, Soudry, and Taylor, "l-adic representations associated to modular forms over imaginary quadratic fields," Inventiones, 1993. The paper is available here: <https://eudml.org/doc/144109>


