If n = 2 k + 1 is odd, then Δ Δ ⊗ ⊗ Δ Δ ∗ ∗ ≅ ≅ ⨁ ⨁ p = 0 k Γ Γ 2 p . In the real case, once again the isomorphism holds Δ Δ ¯ ¯ ≅ ≅ σ σ − − Δ Δ ∗ ∗ . Hence there is a Clebsch–Gordan decomposition (again using the Hodge star to dualize) given by Δ Δ ⊗ ⊗ Δ Δ ¯ ¯ ≅ ≅ σ σ − − Γ Γ 0 ⊕ ⊕ σ σ + Γ Γ 1 ⊕ ⊕ ⋯ ⋯ ⊕ ⊕ σ σ ± ± Γ Γ k