Algebraic groups of multiplicative type in char 0

Question

For a number field $k$ (so of char 0), are algebraic $k$-groups of multiplicative type always linear?

Answer 1

I assume that this is the result in SGA 3 that Scott Carnahan references, but the following is true:

Suppose that we have a fibered diagram as follows $$\begin{matrix}X & \to & Y\\ \downarrow & & \downarrow\\ T & \to & S\end{matrix}$$ where $T\to S$ is fpqc. Then, $Y\to S$ is affine, if and only if $X\to T$ is affine.

So, in your problem, if you start with an algebraic group $G/\text{Spec}(k)$, and the base change to $\text{Spec}(\overline{k})$, is affine, then the above implies that $G$ is affine.