group representations follow from . It follows that the irreducible representations have real matrix representatives if and only if . Reducible representations on the form have real matrices too. The adjoint representation, the Clifford algebra, and the Dirac spinor representation In general representation theory, if is a representation of a Lie algebra then there is an associated representation of on , also denoted , given by Likewise, a representation of a group yields a representation on of , still denoted , given by If and are the standard representations on and if the action is restricted to then the two above representations are the adjoint representation of the Lie algebra and the adjoint representation of the group respectively. The corresponding representations (some or ) always exist for any matrix Lie group, and are paramount for investigation of the representation theory in general, and for any given Lie group in particular. Applying this to the Lorentz group, if is a projective representation, then direct calculation using shows that the induced representation on is a proper representation, i.e. a representation without phase factors. In quantum mechanics this means that if or is a representation acting on some Hilbert space , then the corresponding induced representation acts on the set of linear operators on . As an example, the induced representation of the projective spin representation on is the non-projective 4-vector (, ) representation. For simplicity, consider only the "discrete part" of , that is, given a basis for , the set of constant matrices of various dimension, including possibly infinite dimensions. 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, whose complexification is 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 spin representation The six-dimensional representation space of the tensor -representation inside has two roles. The where