Unipotent Group Scheme

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Can someone give me a reference of why this is true?

Let A an associative k-algebra in wich every element is nilpotent, (non unital) and free of finite rank k-module, then for every commutative unital k-algebra R, we have the group $(A \otimes_k R)^*$ where $(A \otimes_k R)^*$ has the same underlying set and the operation $xºy=x+y+xy$.

So the functor $Spec\;R \rightarrow (A \otimes_k R)^*$ is representable by a unipotent group scheme over k.

My problem is in the second part. And any intuitive way to see it it would help too.