  We define a class of groups constructed from rings equipped with an
involution. We show that under suitable conditions, these groups are either
algebraic or arithmetic, including as special cases the orientation-preserving
isometry group of hyperbolic 4-space, $SL(2,R)$ for any commutative ring $R$,
various symplectic and orthogonal groups, and an important class of arithmetic
subgroups of $SO^+(4,1)$. We investigate when such groups are isomorphic and
conjugate, and relate this to problem of determining when hyperbolic
$4$-orbifolds are homotopic.
