I have recently been inspecting the Lie group of geometric transformations generated by the "null polarities" in real projective 3-space associated with nonsingular, nondegenerate "linear complexes" [sets of lines subject to single linear constraints on their Pluecker coordinates]. Amongst other things, I have ascertained that: This group is isomorphic to the indefinite orthogonal group O(3,3) [preserving the quadratic form (x1)^2+(x2)^2+(x3)^2-(y1)^2-(y2)^2-(y3)^2] over the real numbers; it has dimension 15. It includes the dimension 6 Euclidean group SE(3) of proper (continuous) isometries; also the dimension 3 "axial dilations" transforming a sphere into an ellipsoid. Are these properties well-known? Are they interesting or useful? What else is known about this group? [I am particularly interested in characterising the continuous component synthetically; as well as establishing its relationship to the other three.] Can anybody direct me to pertinant existing references? Fred Lunnon [23/09/07]