Singularity of the generator subalgebra in mixed $q$-Gaussian algebras

We prove that for the mixed $q$-Gaussian algebra ${\mathrm{\Gamma }}_Q(H_{\mathbb{R}})$ associated to a real Hilbert space $H_{\mathbb{R}}$ and a real symmetric matrix $Q=(q_{ij})$ with $\sup \mathrm{\mid }{q}_{ij}\mathrm{\mid }<1$, the generator subalgebra is singular.