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As discussed in a previous article, any (real) Lorentz algebra elementpossess a unique orthogonal decomposition as a sum of two mutually annihilatingdecomposable Lorentz algebra elements. In this article, this concept isextended to the spin representation of the Lorentz algebra. As an application,a formula for the exponential of the spin representation is obtained, as wellas a formula for the spin representation of a proper orthochronous Lorentztransformation.

The exponential of the spin representation of the Lorentz algebra