F-split $SO(2)$ is isomorphic to $GL(1)$?

Question

Let $F$ be a number field and $\chi$ is a character of $\mathbb{A}^{\times}/F^{\times}$.

In some paper, it is written that we can consider $\chi$ as a representation of $F$-split $SO(2)$.

I am wondering whether $F$-split $SO(2)$ isomorphic to $GL(1)$?

Thanks in advance!