Let $G$ be a complex reductive group with Lie algebra $\mathfrak{g}$, and let $(g,X)\in G\times\mathfrak{g}^{\operatorname{reg}}$ be such that $\mathrm{Ad}_gX=X$.
I am wondering if the following identity holds.
$$(1-\mathrm{Ad}_g)(\mathfrak{g})\subseteq\mathrm{ad}(X)(\mathfrak{g})$$
I believe it is true when $G=\mathrm{SL}(2,\mathbb{C})$, as I verified it by brut force. I also believe it is true for all $G=\mathrm{SL}(n,\mathbb{C})$, as I verified the identity in many cases using a symbolic mathematical computation program. But I don't have any conceptual reason for why this happens.
More generally, my question is if we can determine a subclass of the complex reductive groups for which the identity holds (for all pairs $(g,X)$).