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

In the present paper we complete the description of the subgroups generated by a pair of $2$-tori in $\mathrm{GL}(n,K)$. Recall that $2$-tori in $\mathrm{GL}(n,K)$ are the subgroups conjugate to the diagonal subgroup of the following form $\mathrm{diag}(\varepsilon, \varepsilon, 1,\dots, 1)$. In work [2] the reduction theorem for the pairs of $m$-tori was proved. It follows from it that any pair of $2$-tori can be embedded in $\mathrm{GL}(6,K)$ by simultaneous conjugation. The orbit of a pair of $2$-tori $(X,Y)$ is called the orbit in $\mathrm{GL}(n,K)$, if the pair $(X,Y)$ is embedded in $\mathrm{GL}(n,K)$ by simultaneous conjugation and it can not be embedded in $\mathrm{GL}(n-1,K)$. It is clear that $n$ can take values $3, 4, 5$ and $6$. In the same work the orbits and spans of $2$-tori in $\mathrm{GL}(6,K)$ were described. In the subsequent papers we described the pairs of $2$-tori in $\mathrm{GL}(5,K)$, the orbits of pairs of $2$-tori in $\mathrm{GL}(4,K)$ and the spans in $\mathrm{GL}(4,K)$ corresponding to degenerate cases (the reductive part of the group is not larger than $\mathrm{GL}(2,K)$). In this paper we describe undegenerate cases of pairs of $2$-tori in $\mathrm{GL}(4,K)$. Thus we complete our description. The most difficult subgroups turns out the groups with a reductive part $\mathrm{SL}(2,K)\times \mathrm{SL}(2,K)$ or $\mathrm{SL}(2,L)$, where $[L:K]=2$.