I'm getting a bit confused about the notation with the exponential map and the adjoint action.
Could someone explain to me what $$ \text{exp}(t\text{ } ad(X))(Y) $$ means, where $X,Y$ are both elements of a Lie algebra? Is this supposed to mean take the $ad(X)(Y) = [X,Y]$ and then multiply by $t$ and exponentiate, or is there something funny I'm missing?
Based on the way the parentheses are placed, it means first take the exponential of $t \operatorname{ad}(X)$, and then apply that to $Y$. If $\frak g$ is the Lie algebra containing $X$ and $Y$, then $t \operatorname{ad}(X)$ is an element of $\frak g\frak l(\frak g)$ (the space of linear endomorphisms of $\frak g$), which is the Lie algebra of $\operatorname{GL}(\frak g)$ (the group of invertible endomorphisms of $\frak g$). Thus $\exp (t \operatorname{ad}(X))\in \operatorname{GL}(\frak g)$, so it makes sense to apply it to $Y\in \frak g$.