On polynomial products in nilpotent and solvable Lie groups

We are dealing with Lie groups $G$ which are diffeomorphic to ${\mathbb R}^n$, for some $n$. After identifying $G$ with ${\mathbb R}^n$, the multiplication on $G$ can be seen as a map $\mu:{\mathbb R}^n\times {\mathbb R}^n\rightarrow{\mathbb R}^n: (\mathbf{x},\mathbf{y})\mapsto \mu(\mathbf{x},\mathbf{y})$. We show that if $\mu$ is a polynomial map in one of the two (sets of) variables $\mathbf{x}$ or $\mathbf{y}$, then $G$ is solvable. Moreover, if one knows that $\mu$ is polynomial in one of the variables, the group $G$ is nilpotent if and only if $\mu$ is polynomial in both its variables.