Involutions on a group of Möbius transformations 1 Möbius transformations

certain restrictions on the entries of A such that ψ(x) = (ax+ b)(cx+ d)−1, x ∈ Rn, and cx+ d 6= 0. Let O(n) be the real orthogonal group, i.e., the group of invertible linear transformations of Rn that preserve the inner-product K. For any v ∈ Rn, let v⊥ = {x ∈ Rn : K(x, v) = 0}, span(v) = {αv : α ∈ R}, and let ‖v‖ = K(v, v)1/2 be the Euclidean norm. Suppose v is a unit vector, i.e., ‖v‖ = 1. Let fv ∈ O(n) satisfy fv(v) = −v, and fv(x) = x whenever x ∈ v⊥. We set fv(∞) = ∞. Since v2 = −1 and xv = −vx if x ∈ v⊥, we obtain fv(x) = vxv for all x ∈ Rn. Applying (1), we obtain a well-known identity