An embedding of the Poincaré Lie algebra into an extension of the Lie field of SO₀(1,4)

The * isomorphism between algebraic extensions of the Lie fields of SO₀(1,4) and the Poincaré group are considered herein. It is shown that the principal series of unitary ray representations of SO ₀(1,4) are associated, via the * isomorphism, with real mass, positive and negative energy representations of the Poincaré Lie algebra with arbitrary spin. Results on the most degenerate exceptional series of SO₀(1,4) representations are also given.