The product of Lindelöf groups and $\mathbb{R}$-factorizability

Lindelöf topological groups $G_1$ , $H_1$, $G_2$, $H_2$ are constructed in such a way that the products of $G_1\times H_1$ and $G_2\times H_2$ are not $\mathbb{R}$-factorizable groups and (1) the group $G_1\times H_1$ is not pseudo-${\mathrm{\aleph }}_1$-compact; (2) the group $G_2\times H_2$ is a separable not normal group and contains a discrete closed subset of the cardinality continuum.