Let $G$ be a reductive group over a field $k$ of characteristic zero. The Jacobson-Morozov theorem gives a method of embedding any unipotent element into an $\mathfrak{sl}_2$ triple, which in turn induces a parabolic subgroup of $G$ given by of $\mathfrak p=\oplus_{n\ge0}\mathfrak g_n$, called the canonical parabolic.

It is known, e.g. from the discussion here, that not all parabolic subgroups of $G$ arise as canonical parabolics as we range over unipotent elements of $G$. But I am wondering if all Levi subgroups of $G$ do arise from the Jacobson-Morozov triples, namely from the collection of all such $\mathfrak g_0$?