When dim(V) is even, V = W ⊕ W ′ where W ′ is the chosen isotropic complement. Hence any v ∈ V decomposes uniquely as v = w + w ′ with w ∈ W and w ′ ∈ W ′. The action of v on a spinor is given by c ( v ) w 1 ∧ ∧ ⋯ ⋯ ∧ ∧ w n = ( ϵ ϵ ( w ) + i ( w ′ ) ) ( w 1 ∧ ∧ ⋯ ⋯ ∧ ∧ w n ) where i (w ′) is interior product with w ′ using the nondegenerate quadratic form to identify V with V, and ε (w) denotes the exterior product. This action is sometimes called the Clifford product. It may be verified that c ( u ) c ( v ) + c ( v ) c ( u ) = 2 g ( u , v ) , and so c respects the Clifford relations and extends to a homomorphism from the Clifford algebra to End(Δ).