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Operadic Lax representations for the harmonic oscillator are used toconstruct the dynamical deformations of three-dimensional (3D) real Liealgebras in the Bianchi classification. It is shown that the energyconservation of the harmonic oscillator is related to the Jacobi identities ofthe dynamically deformed algebras. Based on this observation, it is proved thatthe dynamical deformations of 3D real Lie algebras in the Bianchiclassification over the harmonic oscillator are Lie algebras.

Dynamical deformations of three-dimensional Lie algebras in Bianchi classification over the harmonic oscillator