  Operadic Lax representations for the harmonic oscillator are used to
construct the quantum counterparts of some 3d real Lie algebras in Bianchi
classification. The Jacobians of these quantum algebras are studied. It is
conjectured that the tangent algebras of these quantum algebras are the
Heisenberg algebra. From this it follows that the volume element in
$\mathbb{R}^{3}$ is quantized by $|(x,y,z)|=4\sqrt{2}(2n+1)$,
($n=0,1,2,\dots$). Thus, the elementary (minimal) length in this model is
$l_{min}=2^{5/6}$.
