Finite dimensional Lie Group with a vector space structure (like diffeomorphisms)

Question

Diffeomorphisms are a infinite dimensional Lie group, but you can also add and scalar multiply them like functions, so they act like a infinite dimensional vector space. Are there any non-trivial (not $\mathbb{R}^n$) finite dimensional Lie groups such that I can define $g_1 + \alpha g_2 \in G$ for $g_1, g_2 \in G, \alpha \in \mathbb{R}$? If there isn't one is there a proof why this can't exist?

Answer 1

I suggest the Heisenberg group $$H= \left\{ \begin{pmatrix} 1 & a &b\\ 0 & 1 &c\\ 0 & 0 &1 \end{pmatrix} :a,b,c \in \mathbb{R} \right\}$$ with the addition $$ \begin{pmatrix} 1 & a &b\\ 0 & 1 &c\\ 0 & 0 &1 \end{pmatrix} + \begin{pmatrix} 1 & d &f\\ 0 & 1 &g\\ 0 & 0 &1 \end{pmatrix} =\begin{pmatrix} 1 & a+d &b+f\\ 0 & 1 &c+g\\ 0 & 0 &1 \end{pmatrix} $$ and scalar multiple $$\alpha \begin{pmatrix} 1 & a &b\\ 0 & 1 &c\\ 0 & 0 &1 \end{pmatrix} =\begin{pmatrix} 1 & \alpha a &\alpha b\\ 0 & 1 &\alpha c\\ 0 & 0 &1 \end{pmatrix}. $$