The induced 4-vector representation of above on this simplified has an invariant 4-dimensional subspace that is spanned by the four gamma matrices. (The metric convention is different in the linked article.) In a corresponding way, the complete Clifford algebra of spacetime, formula_200 whose complexification is formula_201 generated by the gamma matrices decomposes as a direct sum of representation spaces of a scalar irreducible representation (irrep), the , a pseudoscalar irrep, also the , but with parity inversion eigenvalue , see the next section below, the already mentioned vector irrep, , a pseudovector irrep, with parity inversion eigenvalue +1 (not −1), and a tensor irrep, . The dimensions add up to . In other words,  where, as is customary, a representation is confused with its representation space. The six-dimensional representation space of the tensor -representation inside formula_202 has two roles. The  where formula_203 are the gamma matrices, the sigmas, only of which are non-zero due to antisymmetry of the bracket, span the tensor representation space. Moreover, they have the commutation relations of the Lorentz Lie algebra,  and hence constitute a representation (in addition to spanning a representation space) sitting inside formula_200 the spin representation. For details, see bispinor and Dirac algebra. The conclusion is that every element of the complexified formula_202 in (i.e. every complex matrix) has well defined Lorentz transformation properties. In addition, it has a spin-representation of the Lorentz Lie algebra, which upon exponentiation becomes a spin representation of the group, acting on formula_206 making it a space of bispinors. There is a multitude of other representations that can be deduced from the irreducible ones, such as those obtained by taking direct sums, tensor products, and quotients of the irreducible representations. Other methods of obtaining representations include the restriction of a representation of a larger group containing the Lorentz group, e.g. formula_207 and the Poincaré group. These representations are in general not irreducible. The Lorentz group and its Lie algebra have the "complete reducibility property". This means that every representation reduces to a direct sum of irreducible representations. The reducible representations will therefore not be discussed.