1. Here a local Lorentz transformation means a field of Lorentz transformations of the tangent space at each point.
2. That is to say, to observers whose velocity vector is tangent to the curves of the congruence.
3. A synchronization is a specification of the locus of points (hypersurfaces) of equal time. Every synchronization has a natural set of observers: those whose velocity vector is normal to the family of hypersurfaces.
4. They are given biunivocally only for exponential groups (J. Dixmier, (1957). Bull. Soc. math. France,85, 113; L. Pukanszky, (1967). Trans. Amer. Math. Soc.,126, 487).
5. Because of the space-time metric, the elements of the algebra may be written as second order antisymmetric covariant tensors at each point so that, in the corresponding domain of the space-time, they define a two-form.