In detail, let V be a finite-dimensional complex vector space with nondegenerate symmetric bilinear form g. The Clifford algebra Cℓ(V, g) is the algebra generated by V along with the anticommutation relation xy + yx = 2 g (x, y). It is an abstract version of the algebra generated by the gamma or Pauli matrices. If V = C n , with the standard form g (x, y) = x y = x 1 y 1 +... + x n y n we denote the Clifford algebra by Cℓ n ( C ). Since by the choice of an orthonormal basis every complex vectorspace with non-degenerate form is isomorphic to this standard example, this notation is abused more generally if dim C (V) = n. If n = 2 k is even, Cℓ n ( C ) is isomorphic as an algebra (in a non-unique way) to the algebra Mat(2, C ) of 2 × 2 complex matrices (by the Artin–Wedderburn theorem and the easy to prove fact that the Clifford algebra is central simple). If n = 2 k + 1 is odd, Cℓ 2 k +1 ( C ) is isomorphic to the algebra Mat(2, C ) ⊕ Mat(2, C ) of two copies of the 2 × 2 complex matrices. Therefore, in either case Cℓ(V, g) has a unique (up to isomorphism) irreducible representation (also called simple Clifford module), commonly denoted by Δ, of dimension 2. Since the Lie algebra so (V, g) is embedded as a Lie subalgebra in Cℓ(V, g) equipped with the Clifford algebra commutator as Lie bracket, the space Δ is also a Lie algebra representation of so (V, g) called a spin representation. If n is odd, this Lie algebra representation is irreducible. If n is even, it splits further into two irreducible representations Δ = Δ + ⊕ Δ − called the Weyl or half-spin representations.