Commutator groups of orthogonal groups over the reals with emphasis on Lorentz groups

Let Omega be the commutator subgroup of the n-dimensional Lorentz group. We give a criterion when an element of Omega is a product of 2 or 3 involutions of Omega. We prove that a real element of Omega is 2-reflectional. Then we study orthogonal groups over the reals with arbitrary signature. In this situation each real element in the kernel of the spinorial norm is 2-reflectional in the kernel of the spinorial norm. A main result states that the commutator group Omega(p, q) of an orthogonal group O(p, q) over the reals is 2-reflectional if and only if the signature (p, q) satisfies p, q, p q not equivalent to 2 mod 4. For all special orthogonal groups (over arbitrary fields) we prove that real elements are 2-reflectional. (C) 2010 Elsevier Inc. All rights reserved.