This question has some overlap with previous ones but doesn't seem to have a well-documented answer. I recall some literature (mostly involving Lie groups and hermitian symmetric pairs, etc.) which concerns *maximal* parabolic subalgebras of a simple Lie algebra $\mathfrak{g}$ over the field $\mathbb{C}$ or related parabolic subgroups of Lie groups. Here $\mathfrak{p}$ can be taken as standard relative to some fixed Borel subalgebra and Cartan subalgebra, with a decomposition as direct sum of a Levi subalgebra (involving all but one simple root) along with the nilradical $\mathfrak{n}$.

In this situation, is there a necessary and sufficient criterion in the literature for $\mathfrak{n}$ to be abelian, which can then be checked easily case-by-case for the simple types?

For example, neither of the two types of maximal (=minimal) parabolic subalgebra in the Lie algebra $G_2$ turns out to have an abelian nilradical. It's also true that $G_2$ has no *minuscule* highest weights for its irreducible finite dimensional representations.

On the other hand, one of the equivalent conditions for a dominant integral weight to be (co)minuscule implies that the nilradical of the parabolic subalgebra stabilizing a highest weight vector in the corresponding representation must be abelian. (See my previous question here for some references on minuscule weights.) But I don't recall now exactly how precise a criterion exists in the literature for $\mathfrak{n}$ to be abelian when $\mathfrak{p}$ is maximal.

[I was just thinking about this in connection with a newer question here which is not precisely formulated but apparently involves a similar setting.]

ADDED: The answer (plus email) and references given are quite helpful though somewhat intertwined with Lie groups and differential geometry or algebraic groups and algebraic geometry. I was looking for a self-contained approach via roots and weights within the traditional Lie algebra setting. Anyway, a uniform elementary statement seems to emerge as follows. With $\mathfrak{g}$ simple, take $\mathfrak{p}$ to be a maximal parabolic corresponding to the set of simple roots excluding $\alpha$. Then $\mathfrak{n}$ is abelian iff $\alpha$ has coefficient 1 in the highest root, or iff $\mathfrak{p}$ is the stabilizer of a highest weight vector in the irreducible representation whose highest weight is "cominuscule" relative to $\alpha$ (involving interchange of types B, C). (These criteria are then easy to check case-by-case.)