Let $E=\mathbb Q(\sqrt{-d})$ be an imaginary quadratic field and let $R_{E/\mathbb Q}(\mathbb G_m)$ be the restriction of scalars of the multiplicative group, i.e. $R_{E/\mathbb Q}(\mathbb G_m)(X) = \mathbb G_m(X \times_{\mathbb Q} E)$ for each $\mathbb Q$-scheme $X$.
Picking a basis $\langle 1, -\sqrt{-d}\rangle$ for $E$ and letting $E$ act on itself, we can embed $E$ into $M_2(\mathbb Q)$ by $(\alpha, \beta) \mapsto \begin{bmatrix}\alpha&-d\beta\\\beta&\alpha\end{bmatrix}$.
Question 1: This embedding should give rise to an embedding of algebraic groups over $\mathbb Q$, $R_{E/\mathbb Q}(\mathbb G_m) \hookrightarrow GL_2$. Is $R_{E/\mathbb Q}(\mathbb G_m)$ a maximal (non-split) torus in $GL_2$? I seem to remember that the elements of maximal non-split tori in $GL_n$ all satisfy $\det = 1$, which is not the case here. What went wrong?
Question 2: Whatever the correct definition of the maximal torus in $GL_2$ obtained from $E$ is, how can we describe its character group explicitly?