Why a nilpotent Lie group must be a matrix group? Simply connected nilpotent Lie groups are linear; $\begin{pmatrix}1&\mathbf R&\mathbf R\\0&1&\mathbf R\\0&0&1\end{pmatrix}/\begin{pmatrix}1&0&\mathbf Z\\0&1&0\\0&0&1\end{pmatrix}$ isn't. See Ado's theorem.