The set of all complex numbers $z$ such that $e^z$ is real, is exactly the set $\mathbb{R}+i\pi \mathbb{Z}$.
My question is : can we have a similar result for matrices ? More precisely : is there a known characterization of the set $\mathcal{S}$ of all $M\in \mathcal{M}_n(\mathbb{C})$, such that $\exp(M)\in GL_n(\mathbb{R})$ ?
Of course $\mathcal{S}$ contains $\mathcal{M}_n(\mathbb{R})+ \mathbb{Z}\pi i I_n$, but is $\mathcal{S}$ actually bigger than this set ? And if yes, what do we know about $\mathcal{S}$ ?