The collection of matrices$$ \left( \begin{array}{lll} a^2 & 2ab &b^2 \\ ac & bc+ad & bd \\ c^2 & 2cd & d^2 \end{array} \right)$$ indexed by $a,b,c,d \in \mathbb{R}$ is a subgroup of $GL(3,\mathbb{R})$ isomorphic to the group $PGL(2,\mathbb{R}).$ This subgroup, denoted $G$, acts by conjugation on $\mathfrak{gl}(3,\mathbb{R}).$
Are there any polynomial functions in the entries of a matrix $ A \in \mathfrak{gl}(3,\mathbb{R})$ invariant with respect to the action by conjugation with elements in $G$ and NOT invariant with respect to the action by conjugation of $GL(3,\mathbb{R})?$
Does the action by conjugation of G has any invariant sub-spaces except its Lie algebra (which is a Lie subalgebra of $\mathfrak{gl}(3,\mathbb{R})$)?