Composition of Lorentz Transformations in Terms of Their Generators

Two-forms in Minkowski space-time may be considered as generators of Lorentz transformations. Here, the covariant and general expression for the composition law (Baker-Campbell-Hausdorff formula) of two Lorentz transformations in terms of their generators is obtained. For simplicity, the expression is first obtained for complex generators, then translated to real ones. Every generator has two essential eigenvalues and two invariant (two-)planes; the eigenvalues and the invariant planes of the Baker-Campbell-Hausdorff composition of two generators are also obtained.