$\DeclareMathOperator\SU{SU}$Let $t_1, t_2, t_3$ be generators of the Lie algebra $\mathfrak{su}(2)$.
Let us consider a Schwartz space $\mathcal{S}$ defined as \begin{equation} \mathcal{S}:= \Bigl\{ \sum_{\alpha=1}^3 F^{(\alpha,j)}t_{\alpha} \text{ }\Bigl \lvert \text{ each } F^{(\alpha,i)} \text{ is a complex-valued Schwartz function for } \alpha=1,2,3 \text{ and } j=1,2, \cdots, n \Big\} \end{equation}
Next, let $g : \mathbb{R}^n \to \SU(2)$ be an arbitrary compactly supported smooth mapping. Then, assuming the adjoint representation of $\SU(2)$ as well as $\mathfrak{su}(2)$, the adjoint action of $\SU(2)$ on $\mathcal{S}$ is defined for any $F \in \mathcal{S}$ by \begin{equation} g \cdot F := g F g^{-1} - i [\nabla g]g^{-1} \end{equation}
Now, my question is that
Is this adjoint action smooth, closed, proper and free?
Smoohness means this action $(g,F) \to g \cdot F$ is a smooth mapping from $C^\infty_c \bigl(\mathbb{R}^n, \SU(2)\bigr) \times \mathcal{S}$ into $\mathcal{S}$.
Closedness means that the orbit map $\pi : \mathcal{S} \to \mathcal{S}/\SU(2)$ is a closed map.
Proper means that for the mapping $\rho : C^\infty_c \bigl(\mathbb{R}^n, \SU(2) \bigr) \times \mathcal{S} \to \mathcal{S} \times \mathcal{S}$ defined by $\rho(g,F) := \bigl( g \cdot F, F \bigr)$, we have $\rho^{-1}(K)$ compact whenever $K$ is compact.
Free means that if $g \cdot F =F$ for some $F \in \mathcal{S}$, then $g = e_{\SU(2)}$.
Here are my thoughts:
If we consider infinite-dimensional notion of smoothness (cf. in the sense of José Luís da Silva), then I think the smoothness requirement should be quite obvious.
Both $\mathcal{S}$ and $C^\infty_c \bigl(\mathbb{R}^n, \SU(2) \bigr)$ are Montel spaces and therefore have the Heine-Borel property. So, I think there is some hope for closedness and properness requirements.
I am not quite sure about if this action is free. It requires that $g F g^{-1} - i [\nabla g]g^{-1} =F$ for some $F \in \mathcal{S}$ leads to $g = e_{\SU(2)}$ identically. However, I am not confident about this...
Could anyone please help me?
