The author introduces a real matrix representation of the Dirac algebra built of \( 4\times 4\) block matrices each element of which belongs to \(\text{ SO}(4)\). The standard space-time metric used is \( \text{ diag}(-1,1,1,1)\). The main purpose is to offer a concrete matrix realization of relevant physical actions and to define generalized Lorentz transformations, that is, transformations of displaced systems, rotating systems, charged systems and others. Poincaré transformations then arise as approximations of the generalized Lorentz transformations. Physical interpretations of the representation in terms of space-time actions are given.