Lemma
Let \(T\) be a rank \(r\) torus over \(k\) a field. Then the character lattice of \(T\) is isomorphic to \(\mathbb Z^r\).
Proof
An algebraic group homomorphism is a function \(\varphi:T\to \mathbb G_m\) which is regular and a group homomorphism. Since both \(T\) and \(\mathbb G_m\) are affine this means \(\varphi\) is globally a polynomial function. The only such polynomials are monomials with coefficient 1. The monomials on \(T\) are then isomorphic to \(\mathbb Z^r\) once you choose a presentation \(k[t_1^{\pm}, ..., t_r^{\pm}]\) for \(\Gamma(T, \mathcal O_T)\); take \(t_i\) to the generator \(e_i\) of \(\mathbb Z^r\), so that
\begin{align*} t_1^{m_1}\cdot\dots\cdott_r^{m_r} \mapsto m_1e_1 + ... + m_re_r. \end{align*}