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 the spin representation. For details, see bispinor and Dirac algebra. The conclusion is that every element of the complexified 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 making it a space of bispinors. Reducible representations 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. 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. Space inversion and time reversal The (possibly projective) representation is irreducible as a representation , the identity component of the Lorentz group, in physics terminology the proper orthochronous Lorentz group. If it can be extended to a representation of all of , the full Lorentz group, including space parity inversion and time reversal. The representations can be extended likewise. Space parity inversion For space parity inversion, the adjoint action of on is considered, where is the standard representative of space parity inversion, , given by It is these properties of and under that motivate the terms vector for and pseudovector or axial vector for . In a similar way, if is any representation of and is its associated group representation, then acts on the representation of by the adjoint action, for . If is to be included in , then consistency with requires that holds, where and are defined as in the first section. This can hold only if and have the same dimensions, i.e. only if . When then can be extended to an irreducible representation of , the orthochronous Lorentz group. The parity reversal representative does not come automatically with the general construction of the