Let $F$ be a commutative field and $n\geqslant 2$ be an integer. It is well known that the maximal anisotropic mod center tori in $G={\rm GL}(n,F)$ are of the form $T = {\rm Res}_{E/F}\; {\mathbb G}_m$,for some degree $n$ separable field extension $E/F$, where ${\rm Res}$ denotes Weil's restriction of scalar and where ${\mathbb G}_m$ denotes the $1$-dimensional split torus.

Such a torus $T$ embeds in $G$ in the following way. One identifies $G$ with ${\rm Aut}_F\; (E)$ and make $E^{\times} =T(F)$ acts on $E$ by multiplication.

One says that a torus is elliptic if it is not contained in any proper parabolic subgroup of $G$. The tori $T$ described above are elliptic.

My question is :

Is it true that the subtori of $T$ that are elliptic and anisotropic are of the form : $S=\{ t\in T\ ; \ N_{E/L}(t)=1\}$, where $L/F$ is a proper subfield extension of $E/F$ ?