Metrics are Clifford algebra involutions

Four-dimensional space-time, all relevant inner products, and some of the groups leaving these inner products invariant are manufactured from more basic algebraic ingredients, all inside the 8-dimensional Pauli algebra : (1) Euclidean 3-spaceE ³, (2) Minkowski 4-spaceM ⁴, (3) complex 4-space ⁴, and all three metrics and all three inner products. The groupsSO(3;) SO(3; 1;) SO (4;) are obtained as images of twofold covering maps of subgroups of or their direct product. A method of embedding in the Clifford algebraC(1;n–1) ofn-dimensional Minkowski space is given for anyn4. Furthermore, all three groups act not only on the relevant vector spaces, but on all ofC(1;n–1), leaving setwise invariant.