Good morning,

I would like to pose the following (maybe naive) question. Let $\mathfrak{a}\subset \mathfrak{gl}(\mathbb{R},d)$ be any lie subalgebra, and $A$ be the connected, simply connected subgroup of $GL(\mathbb{R},d)$ generated by $\mathfrak{a}$.

Assume that for every $x\in\mathbb{R}^d\setminus \{0\}$, the orbit $Ax$ is dense in $\mathbb{R}^d$. Is it true that necessarily there has to be a matrix in $\mathfrak{a}$ which has a nonzero real eigenvalue?

Intuitively I am tempted to say so, since this eigenvalue would cover the ``radial'' part of any such orbit, but I have not been able to come across any similar results in the literature. I have looked for results in representation theory and in the theory of dynamical systems (I have looked for minimal actions of groups), but up to now I have found very scarce references.

Do you know any good reference to have a look at for such kind of questions? Or is my intuition just plainly false? Any counterexample? I'm a bit lost, so any help would be greatly appreciated.

Regards.