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When is the image of the adjoint representation of a real algebraic group Zariski closed?

Let $\operatorname{Ad}:\operatorname{SL}_n(\mathbb{R}) \to \operatorname{GL}(\mathfrak{sl}_n(\mathbb{R}))$ be the adjoint representation (i.e. $\operatorname{Ad}(g)X=gXg^{-1}$) of $SL_n(\mathbb{R})$. …