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For some research questions that involve Spin(p, q) representation theory, using symbolic algebra based techniques might be an attractive option for simplifying and manipulating expressions. Yet, for some such problems, especially as they arise in the study of various limits of M-theory (such as dimensional reductions), the complexity of the resulting expressions can become computationally challenging when using popular symbolic algebra packages in a straightforward manner. This work discusses some general properties of Gamma matrices that are computationally useful, down to the level of what one would call "bit-twiddling hacks" in computer science. It is presented in a self-contained way that should be accessible to both physicists and computer scientists. Code is available alongside the TeX source of the preprint version of this article on arXiv. [ABSTRACT FROM AUTHOR]