Let $k$ be a number field, $l/k$ be a finite extension, and $T_{/l}$ be a linear algebraic torus over $l$.
Is $R_{l/k}(T_{/l})$ a linear algebraic torus over $k$? Here $R_{l/k}$ is the restriction of scalars.
What about the same question above with "group of multiplicative type" instead of "torus"?
Yes. Let's call the torus $T$. Then by definition $R_{l/k}(T)(k_s) = T(l\otimes_k k_s)$ where $k_s$ is the separable (or use algebraic closure, doesn't matter), and $l\otimes_kk_s$ is a product of copies of $k_s$ since $l/k$ is separable. So over the separable closure $R_{l/k}(T)$ still is isomorphic to a bunch of copies of $\mathbb{G}_m$. One can also see that this applies to diagonalizable subgroups.