Let $\xi \in T_xOx$ be a tangent vector at $x \in O_x :=\{\mathrm{ Ad} _{u}^*(x); u \in G\}$ for $x \in g^*.$ ($g$ is the Lie-Algebra)
Then I read that this $\xi$ can be represented as the velocity vector of a coadjoint action of a one-parameter subgroup $e^{ta}$ for $a \in g.$
In particular
$\xi = \frac{d}{dt}|_{t=0} \mathrm{Ad} ^*_{e^{ta}}(x) = \mathrm{ad} _a^*(x).$
The thing is that I don't understand why such an $a \in g$ must exist. Could anybody explain me why this is the case?