Let $G$ be an affine reductive group defined over a field of characteristic zero $k$, denote by $\bar{k}$ the algebraic closure of $k$, and by $k^{\times}$ the multiplicative group of $k$. Let $Z(G)$ be the center of $G$. Let $\lambda$ be a 1parameter subgroup of $G$ defined over $k$. Suppose that $\lambda(k^{\times})\subset Z(G)$, is it true that $\lambda(\bar{k}^{\times})\subset Z(G)$ or can you give me a counterexample?
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