We have a matrix C such that the map c(x) = Cx fixes Z(0,0,1), and has the property that CTKC = K, where K = diag(1,1,-1). Since c maps Z to Z, the third column of C must be (0,0,1)T. We partition the matrices into the top left-hand 2x2 matrix, and 2x1, 1x2 and 1x1 blocks. K = |I 0| |0T -1| C = |U O| |uT 1| where 0 is the zero vector in R2, u is some vector in R2, and U is a 2x2 matrix. A direct calculation shows that CTKC = |UTU-uuT -u| |-uT -1| But this is K, so u = 0, and hence UTU = I, so that U is orthogonal, and so U acts on R2 as a rotation about O, or reflection in some line through O. It follows that C acts on R3 is a rotation about the z-axis or a reflection in a plane through the z-axis.

the hyperbolic-projective theorem