Let $G$ be a group scheme over $S$ where $S$ is a reduced scheme of finite type over a field $k$ of characteristic 0, and let every fibre $G_s$ over a closed point of $S$ be isomorphic to $\mathbb{G}_a^n$ for some $n$ that varies with $s$.

<s>Is it true then that $G$ is reduced?</s>

Suppose that $G$ is reduced. Is it true that $G \times_S G$ is reduced?

(This question is a continuation of [this one](http://mathoverflow.net/questions/88360/is-tensor-square-of-a-reduced-ring-reduced); the motivation comes from [this question](http://mathoverflow.net/questions/88108/reduction-of-a-module))