The Poincaré algebra is the Lie algebra of the Poincaré group. It is a Lie algebra extension of the Lie algebra of the Lorentz group. More specifically, the proper ( det Λ Λ = 1 ), orthochronous ( Λ Λ 0 0 ≥ ≥ 1 ) part of the Lorentz subgroup (its identity component), S O ( 1 , 3 ) + ↑ ↑ , is connected to the identity and is thus provided by the exponentiation exp ⁡ ⁡ ( i a μ μ P μ μ ) exp ⁡ ⁡ ( i 2 ω ω μ μ ν ν M μ μ ν ν ) of this Lie algebra. In component form, the Poincaré algebra is given by the commutation relations: