Note that for n even, this computation also shows that Δ Δ = C ℓ ℓ ( W ) ω ω = ( Λ Λ ∗ ∗ W ) ω ω as a vector space. In the last equality we again used that W is isotropic. In physics terms, this shows that Δ is built up like a Fock space by creating spinors using anti-commuting creation operators in W acting on a vacuum ω.