Example
Let S be the restriction of scalars of Gm over the field extension C/R. This is a real torus whose real points form the Lie group of nonzero complex numbers. Restriction of scalars gives a canonical embedding of S into GL2, and composition with determinant gives an algebraic homomorphism of tori from S to Gm, called the norm. The kernel of this map is a nonsplit rank one torus called the norm torus of the extension C/R, and its real points form the Lie group U(1), which is topologically a circle. It has no multiplicative subgroups (equivalently, the weight lattice has no nonzero Galois fixed points), and such tori are called anisotropic. Its weight lattice is a copy of the integers, with the nontrivial Galois action that sends complex conjugation to the minus one map.
Read more about this topic: Algebraic Torus
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