Quantization of $c$-number Dirac-Bracket

I have a question concerning the quantization of phase-space variables $(q_1, q_2, q_3, p_1, p_2, p_3)$ with the Hamiltonian

$$
H = frac{3}{2}(p_1^2+p_2^2 +p_3^2)
$$

and the following non-commuting second class constraints:

$$
Phi_1 = q_1+q_2+q_3=0
Phi_2 = p_1+p_2+p_3=0.
$$

The general method proposed by Dirac in such a case is to compute the Dirac Bracket

$$
[F,G]_D = [F,G] – [F, Phi_i]c^{ij}[Phi_j,G]
$$

where $c^{ij} = [Phi_i,Phi_j]^{-1}$ and $[cdot, cdot]$ is the usual Poisson bracket. In this case one has to deal with c-number Dirac brackets, i.e brackets which amount to a complex/real number:

$$
[q_i,q_j]_D = 0= [p_i,p_j]_D
[q_i,p_j]_D = delta_{ij} – frac{1}{3}.
$$

Quoting Henneaux & Teitelboim (https://press.princeton.edu/books/paperback/9780691037691/quantization-of-gauge-systems), Chapter 13, page 273ff, it should be possible to quantize such a structure. Unfortunately, they never state how exactly one should approach such a task. I experimented with different combination of first-order differential operators and coordinates, similiar to the classical Poisson-bracket, but to no avail. Is there a general recipe how this is done?

Caveat: If feasible, the whole algebra should be quantized. I know it is possible to find new (Darboux) coordinates on the constraint surface, express a new bracket and quantize it. But this is not the primary goal here.