Commutator groups of orthogonal groups over the reals with emphasis on Lorentz groups. (Q603140)

Let \(\Omega\) be the commutator subgroup of the \(n\)-dimensional Lorentz group. A criterion is given for an element of \(\Omega\) to be a product of 2 or 3 involutions in \(\Omega\). Any real element of \(\Omega\) is 2-reflectional. Then orthogonal groups over the reals with arbitrary signature are studied. 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 \(p,q,p+q \not\equiv 2\bmod 4\). For all special orthogonal groups (over arbitrary fields), the real elements are 2-reflectional. (For any group, an element is called `real' if it is similar to its inverse.)