Clifford algebras for SE(3) and the Poincaré group

Non-homogeneous groups like the special Euclidean group SE(3) (of the translations and rotations in R 3) or the Poincaré group of Minkowski space-time R 4 also contain translations, which seems to preclude the possibility of building a representation based on Clifford algebra for them because representations are purely based on matrix multiplication. Summing of group elements is a priori not defined, only products are. However, with the use of homogeneous coordinates, SE(3) can be obtained as the group generated by an even number of reflections with respect to hyperplanes of R 4. Similarly the Poincaré group is a group that can be generated by an even number of reflections with respect to hyperplanes of R 5. However, this leads to some normalization problems and in order to avoid these, null vectors must be introduced. We work this out for the group SE(3) and for the Poincaré group. It leads to groups noted as SO(3,1,0) for SE(3) and SO(3,1,1) or SO(1,1,3) for the Poincaré group.