The first orthogonality relation is ∑ ∑ J = | j 1 − − j 2 | j 1 + j 2 ∑ ∑ M = − − J J ⟨ ⟨ j 1 m 1 j 2 m 2 | J M ⟩ ⟩ ⟨ ⟨ J M | j 1 m 1 ′ j 2 m 2 ′ ⟩ ⟩ = ⟨ ⟨ j 1 m 1 j 2 m 2 | j 1 m 1 ′ j 2 m 2 ′ ⟩ ⟩ = δ δ m 1 , m 1 ′ δ δ m 2 , m 2 ′ (derived from the fact that 1 = ∑ ∑ x | x ⟩ ⟩ ⟨ ⟨ x | ) and the second one is ∑ ∑ m 1 , m 2 ⟨ ⟨ J M | j 1 m 1 j 2 m 2 ⟩ ⟩ ⟨ ⟨ j 1 m 1 j 2 m 2 | J ′ M ′ ⟩ ⟩ = ⟨ ⟨ J M | J ′ M ′ ⟩ ⟩ = δ δ J , J ′ δ δ M , M ′ .