It is equipped to handle certain non-commutative algebras which are extensively used in theoretical high energy physics: Clifford algebras, SU(3) Lie algebras, and Lorentz tensors.
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Nevertheless, particles can indeed be neatly divided into groups that form irreducible representations of the Lie algebra SU(3), as first noted by Murray Gell-Mann and independently by Yuval Ne'eman (see the eightfold way).