Example for a linear algebraic $\mathbb{C}$-group not defined over $\mathbb{Q}$? Let $G$ be a linear algebraic group, embedded as a Zariski-closed subgroup of $GL(n,\mathbb{C})$ with respect to the identification of $GL(n,\mathbb{C})$ as a subset of $\mathbb{C}^{n^2+1}$ via $M \mapsto (M,\det(M)^{-1})$.
Then $G$ is said to be defined over $\mathbb{Q}$ if it is the vanishing set of polynomials $f_1,\ldots,f_m \in \mathbb{Q}[X_{11},\ldots,X_{nn},T]$.
All linear algebraic groups I know of are defined over $\mathbb{Q}$, for example $GL(n), SL(n), SO(n)$, invertible diagonal matrices $T_n$, $\mathbb{C}^\times, (\mathbb{C}^n,+)$ etc.
What is an instructive example of a group not defined over $\mathbb{Q}$?

How about the group of matrices of the form $\begin{pmatrix} 1 & x & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \pi x \\ 0 & 0 & 0 & 1 \end{pmatrix}$ for some $x \in \mathbf{C}$? It's obviously isomorphic to a group defined over $\mathbf{Q}$ -- the additive group -- but it's embedded into $\mathrm{GL}(4)$ in a weird, non-rational way.
