Dirac-like linearisation of  x²+p² with noncommuting position and momentum variables leads to the representation of the standard U(1)\otimes SU(3) symmetry of the three-dimensional harmonic oscillator in the relevant Clifford algebra and the emergence of a formula which we previously proposed to identify with the Gell-Mann--Nishijima--Glashow relation between charge, third component of weak isospin and weak hypercharge. This matrix representation exhibits features not present in the standard treatment of harmonic oscillator. We show that these features, strictly corresponding to the structure of a single quark--lepton generation in the Standard Model, may be understood from the point of view of specific O(6) phase-space transformations, which go beyond U(1)\otimes SU(3), and modify standard canonical commutation relations. It is demonstrated that the whole structure of a single quark--lepton generation corresponds to assuming that the imaginary unit appearing in the canonical commutation relations may acquire an additional ``+'' or ``-'' sign separately for each of the three directions.

PACS numbers: 11.30.--j, 03.65.--w, 02.40.--k