I have this question about closed subgroups of $\mathrm{SL}(n,\mathbb{R})$. So assume I have $H$ a (strict) closed subgroup of $\mathrm{SL}(n,\mathbb{R})$. It is therefore a Lie subgroup of $\mathrm{SL}(n,\mathbb{R})$. My question is: how wild can $H$ be? Is it necessarily contained in a simple subgroup? More precisely, is it possible that
1) H does not preserve a strict subverter space of $\mathbb{R}^n$,
2) H does not preserve a quadratic form,
3) H does not preserve a symplectic form,
etc etc...
or can one always find a simple property that $H$ must satisfy?
The question is rather philosophical so feel free to comment as much as you want.