In her notes
http://www.math.toronto.edu/fiona/courses/algp.pdf
on page 383, Example 4.2 Fiona claims that the group
$$ T = \left\lbrace \pmatrix{ a & b \\ -b & a } \bigg|\, a,b \in \mathbb{C},\, a^2 + b^2 \neq 0 \right\rbrace $$
is a Torus. Furthermore, its an $\mathbb{R}-$ torus which is anisotropic over $\mathbb{R}$.
While it is easy to see that it is defined over $\mathbb{R}$, I cannot see why this is a torus or why it is $\mathbb{R}-$ anisotropic.
Any ideas about this.
This group is isomorphic to $\Bbb C^{*2}$ via $(a,b) \mapsto (a+ib,a-ib)$ so it is a torus.
However it is not anisotropic over $\Bbb R$ because it has characters defined over $\Bbb R$, such as $(a,b) \mapsto a^2+b^2$.