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abelian absolute parallelism acts irreducibly alge algebra homomorphism algebra structure algebraic geometry algebraic variety associative algebra automorphisms basis called Cayley algebra chapter Clifford algebra coefficients commutative complex matrices complex numbers composition algebra constructed course defined Definition denote dimension Dirac matrices direct sum division algebra equations equivalent example Exercise finite dimensional ground field Hence idempotent idempotent decomposition implies integer irreducible representation isomorphism K-vector kernel left ideal Lie group Lie subalgebra linear map linear representation linear subspace linear systems Lorentz group manifold mathematics minimal left ideal multiplication nilpotent notation one-one orbit space physics polynomial properties prove PS(W quaternions quotient rational function rational map real numbers real vector space relation Remark result satisfied scalar field semisimple Show Similiarly simple algebra strict degree subgroup subset Suppose symmetric bilinear form systems theory tensor product Theorem 4.1 unit element vector fields vector product vector space vlv2 Zariski zero