Let $E/F$ be a finite separable extension of a commutative field $F$. Let $T$ be the torus ${\rm Res}_{E/F}\; {\mathbb G}_m$, where ${\rm Res}$ is Weil's restriction of scalar. Is it true that any subtorus of $T$ is of one of the following forms :

1. ${\rm Res}_{L/F}\; {\mathbb G}_m$, where $L/F$ is a subextension of $E/F$;

2. $[{\rm Res}_{L/F}\; {\mathbb G}_m ]^1$, where $L/F$ is as in 1) and where the exponent $1$ means the norm $1$ elements for the extension $L/K$, where $K/F$ is a subextension of $L/ F$ ?

If the answer if "yes", do you have a proof or a reference ?

Let $E/F$ be a finite separable extension of a commutative field $F$. Let $T$ be the torus ${\rm Res}_{E/F}\; {\mathbb G}_m$, where ${\rm Res}$ is Weil's restriction of scalar. Is there a simple description of all subtori of $T$ ?