There is a natural identification of Δ with its contragredient representation via the conjugation in the Clifford algebra: ( α α ω ω ) ∗ ∗ = ω ω ( α α ∗ ∗ ) . So Δ ⊗ Δ also decomposes in the above manner. Furthermore, under the even Clifford algebra, the half-spin representations decompose Δ Δ + ⊗ ⊗ Δ Δ + ∗ ∗ ≅ ≅ Δ Δ − − ⊗ ⊗ Δ Δ − − ∗ ∗ ≅ ≅ ⨁ ⨁ p = 0 k Γ Γ 2 p Δ Δ + ⊗ ⊗ Δ Δ − − ∗ ∗ ≅ ≅ Δ Δ − − ⊗ ⊗ Δ Δ + ∗ ∗ ≅ ≅ ⨁ ⨁ p = 0 k − − 1 Γ Γ 2 p + 1