  In this paper a finite dimensional unital associative algebra is presented,
and its group of algebra automorphisms is detailed. The studied algebra can
physically be understood as the creation operator algebra in a formal quantum
field theory at fixed momentum for a spin 1/2 particle along with its
antiparticle. It is shown that the essential part of the corresponding
automorphism group can naturally be related to the conformal Lorentz group. In
addition, the non-semisimple part of the automorphism group can be understood
as "dressing" of the pure one-particle states. The studied mathematical
structure may help in constructing quantum field theories in a non-perturbative
manner. In addition, it provides a simple example of circumventing
Coleman-Mandula theorem using non-semisimple groups, without SUSY.
