Note that any element of I must have the form αω, by virtue of our assumption that I ⊂ Cℓ(V, g) ω. Let αω ∈ I be any such element. Using the chosen basis, we may write α α = ∑ ∑ i 1 < i 2 < ⋯ ⋯ < i p a i 1 … … i p w i 1 ⋯ ⋯ w i p + ∑ ∑ j B j w j ′ where the a i 1... i p are scalars, and the B j are auxiliary elements of the Clifford algebra. Observe now that the product α α ω ω = ∑ ∑ i 1 < i 2 < ⋯ ⋯ < i p a i 1 … … i p w i 1 ⋯ ⋯ w i p ω ω . Pick any nonzero monomial a in the expansion of α with maximal homogeneous degree in the elements w i : a = a i 1 … … i max w i 1 … … w i max (no summation implied), then w i max ′ ⋯ ⋯ w i 1 ′ α α ω ω = a i 1 … … i max ω ω is a nonzero scalar multiple of ω, as required.