Let $L$ be a finite-dimensional Lie algebra over an algebraically closed field $k$ of characteristic zero. Let $\operatorname{Int}(L)$ be the adjoint algebraic group, defined as the smallest algebraic subgroup of $\operatorname{Aut}_k(L)$ containing every one-parameter subgroup $t \mapsto \exp(t\operatorname{ad}_L z)$ with $z \in L$ and $\operatorname{ad}_L z$ nilpotent. Any two Cartan subalgebras of $L$ are conjugate under $\operatorname{Int}(L)$.