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Operadic Lax representations for the harmonic oscillator are used toconstruct the quantum counterparts of some 3d real Lie algebras in Bianchiclassification. The Jacobians of these quantum algebras are studied. It isconjectured that the tangent algebras of these quantum algebras are theHeisenberg 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$l_{min}=2^{5/6}$.

VII$^{\hbar}_a$, III$_{a=1}^{\hbar}$, VI$_{a\neq1}^{\hbar}$