On characters of unipotent groups

Question

I was thinking about characters of unipotent groups, this is my question. Let's define a unipotent group as an algebraic linear group formed by unipotent elements, where an element is called unipotent if it acts unipotently on some regular representation. Let $U$ be a unipotent group and $g:U \rightarrow \mathbb{C}^\times$ a character of $U$. Since image of groups under homomorphisms are groups and the Jordan decomposition is preserved, we have that $g(U) \subset \mathbb{C}^\times$ is a unipotent subgroup. But, the only unipotent subgroup of $\mathbb{C}^\times$ is $\{1\}$. Therefore, we have only the trivial character. Is this argument right?