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