Elementary transvections in the overgroups of a non-split maximal torus

A subgroup $H$ of the general linear group $GL(n, k)$ is rich in transvections if $H$ contains elementary transvections $t_{ij}(\alpha)$ at all positions $(i, j)$, $i\neq j$. In this paper we show that if a subgroup $H$ contains a non-split maximal torus and elementary transvection in one position, than $H$ is rich in transvections. It is also proved that if a subgroup $H$ contains a cyclic permutation of order $n$ and elementary transvection at position $(i, j)$ such that numbers $i-j$ and $n$ are coprime, then $H$ is rich in transvections.