Subgroups generated by a pair of $2$-tori in $\mathrm{GL}(4,K)$. I

This paper is the third one in the series of the works dedicated to the geometry of $2$-tori, i.e. subgroups conjugate to the diagonal subgroup of the form $\big\{\mathrm{diag}\,(\varepsilon,\varepsilon,1,\ldots,1), \varepsilon\in K^*\big\}$, in the general linear group $\mathrm{GL}(n,K)$ over the field $K$. In the first one we proved a reduction theorem establishing that a pair of $2$-tori is conjugate to such a pair in $\mathrm{GL}(6,K)$, and classified such pairs that cannot be embedded in $\mathrm{GL}(5,K)$. In the second we describe the orbits and spans of $2$-tori in $\mathrm{GL}(5,K)$, that cannot be embedded in $\mathrm{GL}(4,K)$. Here we consider the most difficult case of $\mathrm{GL}(4,K)$ and classify the orbits of $\mathrm{GL}(4,K)$ acting by simultaneous conjugation on pairs of $2$-tori.