Consequently, the above inequality applied to this commutation relation specifies Δ Δ L x Δ Δ L y ≥ ≥ 1 2 ℏ ℏ 2 | ⟨ ⟨ L z ⟩ ⟩ | 2 , hence | ⟨ ⟨ L x 2 ⟩ ⟩ ⟨ ⟨ L y 2 ⟩ ⟩ | ≥ ≥ ℏ ℏ 2 2 | m | and therefore ℓ ℓ ( ℓ ℓ + 1 ) − − m 2 ≥ ≥ | m | , so, then, it yields useful constraints such as a lower bound on the Casimir invariant : ℓ (ℓ + 1) ≥ | m | (| m | + 1), and hence ℓ ≥ | m |, among others.