definition
admitted
Split Torus
A torus \(T\) over \(k\) is split if \(T \cong \mathbb{G}_{m,k}^{\,n}\) over \(k\) itself. Equivalently, the \(\Theta\)-action on \(X^*(T)\) (equivalently on \(X_*(T)\)) is trivial; equivalently, every character of \(T_{k_s}\) is defined over \(k\).
For a split torus the character and cocharacter lattices are plain lattices \(\mathbb{Z}^n\) with trivial Galois action, and \(T(k) \cong (k^\times)^n\); more canonically \(T(k) = X_*(T) \otimes_{\mathbb{Z}} k^\times\) via \(\lambda \otimes x \mapsto \lambda(x)\).