Applying the total angular momentum raising and lowering operators J ± ± = j ± ± ⊗ ⊗ 1 + 1 ⊗ ⊗ j ± ± to the left hand side of the defining equation gives J ± ± | [ j 1 j 2 ] J M ⟩ ⟩ = ℏ ℏ C ± ± ( J , M ) | [ j 1 j 2 ] J ( M ± ± 1 ) ⟩ ⟩ = ℏ ℏ C ± ± ( J , M ) ∑ ∑ m 1 , m 2 | j 1 m 1 j 2 m 2 ⟩ ⟩ ⟨ ⟨ j 1 m 1 j 2 m 2 | J ( M ± ± 1 ) ⟩ ⟩ Applying the same operators to the right hand side gives J ± ± ∑ ∑ m 1 , m 2 | j 1 m 1 j 2 m 2 ⟩ ⟩ ⟨ ⟨ j 1 m 1 j 2 m 2 | J M ⟩ ⟩ = ℏ ℏ ∑ ∑ m 1 , m 2 ( C ± ± ( j 1 , m 1 ) | j 1 ( m 1 ± ± 1 ) j 2 m 2 ⟩ ⟩ + C ± ± ( j 2 , m 2 ) | j 1 m 1 j 2 ( m 2 ± ± 1 ) ⟩ ⟩ ) ⟨ ⟨ j 1 m 1 j 2 m 2 | J M ⟩ ⟩ = ℏ ℏ ∑ ∑ m 1 , m 2 | j 1 m 1 j 2 m 2 ⟩ ⟩ ( C ± ± ( j 1 , m 1 ∓ ∓ 1 ) ⟨ ⟨ j 1 ( m 1 ∓ ∓ 1 ) j 2 m 2 | J M ⟩ ⟩ + C ± ± ( j 2 , m 2 ∓ ∓ 1 ) ⟨ ⟨ j 1 m 1 j 2 ( m 2 ∓ ∓ 1 ) | J M ⟩ ⟩ ) .