Irreducibility of the Koopman representations for the group ${\mathrm{G}\mathrm{L}}_0(2\mathrm{\infty },\mathbb{R})$ acting on three infinite rows

Consider the inductive limit of the general linear groups ${\mathrm{G}\mathrm{L}}_0(2\mathrm{\infty },\mathbb{R})$ $={\mathrm{}}_n\mathrm{G}\mathrm{L}(2n-1,\mathbb{R})$, acting on the space $X_m$ of $m$ rows, infinite in both directions, with Gaussian measure. This measure is the infinite tensor product of one-dimensional arbitrary Gaussian non-centered measures. In this article we prove an irreducibility criterion for $m=3$. In 2019, the first author [28] established a criterion for $m\le 2$. Our proof is in the same spirit, but the details are far more involved.