In Lie theory, one often asks about alternating forms on $\mathbb{R}^n$ which are invariant under some particular subgroup $G\subseteq GL_n(\mathbb{R})$, and there is always some algebra of invariant forms associated to $G$. For example, $SO(n)$ preserves the algebra generated by $1\in \Lambda^0(\mathbb{R}^n)$ and $*1\in \Lambda^n(\mathbb{R}^n)$ (and nothing else).

However, taking the algebra and calculating the group which leaves it invariant may yield a strictly larger group than the original one. Of course, another way of looking at it is that sometimes a subgroup of $G$ will still have the same algebra of invariant forms. I've been told, although I have no concrete examples, that these subgroups may not even be nested.

So, I'm wondering if there is some nice algebraic conditions that govern this correspondence. It would be nice to have some characterization along the lines of the basic facts in algebraic geometry that for any subset $T\subseteq A$, $Z(T)=Z((T))$, that for any ideal $a\subseteq A$, $I(Z(a))=\sqrt{a}$, etc.