A quadratic norm-one torus and its sign action

The displayed multiplication is multiplication of $x+y\sqrt{d}$; the norm $x^2-d{y}^2$ multiplies, so it preserves the equation. The identity is $(1,0)$ and the inverse is $(x,-y)$, and associative multiplication follows by direct expansion in the basis $1,\sqrt{d}$. This gives an affine group by F1. Over $L=k(\sqrt{d})$ put $t=x+\sqrt{d}y$. Then $t^{-1}=x-\sqrt{d}y$, and $x=(t+t^{-1})\mathrm{/}2$, $y=(t-t^{-1})\mathrm{/}(2\sqrt{d})$ give inverse algebra maps with $L[t,t^{-1}]$. They preserve multiplication, so $T_L\cong {\mathbf{G}}_m$.

The nontrivial $\sigma \in \mathrm{Gal}(L\mathrm{/}k)$ sends $t$ to $t^{-1}$, so it sends $t^n$ to $t^{-n}$. F2 gives the sign action on $\mathbb{Z}$; F3 makes $T$ a torus. It cannot be split: a split rank-one torus has trivial action, and no isomorphism of abelian groups $\mathbb{Z}\to \mathbb{Z}$ intertwines the sign action with the trivial action (its nonzero generator would have to equal its negative). For $d=-1$ over $\mathbb{R}$ the equation and product are exactly those of unit complex numbers.