TITLE: How can the generators of subalgebra $\mathfrak g^{\sigma}$ of $\sigma$-stable elements be expressed through generators of Lie algebra $\mathfrak g$?
QUESTION [5 upvotes]: Let $\mathfrak g$ be the semisimple Lie algebra of type $D_{4}$. Let $\sigma$ be the 3-rd order automorphism of $\mathfrak g$ induced by the triality of $D_{4}$:

$$
\sigma:\alpha_{1}\mapsto\alpha_{3}\mapsto\alpha_{4},\alpha_{2}\mapsto\alpha_{2}
$$
Let $\mathfrak g^{\sigma}$ be the subalgebra of $\sigma$-stable elements of $\mathfrak g$.
How can Serre generators of $\mathfrak g^{\sigma}$ be expressed through Serre generators of $\mathfrak g$?
I suppose that "Serre generators", "Serre-Chevalley generators", and "Chevalley generators" are equivalent terms. (In my opinion, perhaps the use of a Chevalley basis would be more straightforward?)

REPLY [5 votes]: In the G$_2$ case, I think that you will find all the information you need in Levasseur, T.; Smith, S. P. Primitive ideals and nilpotent orbits in type G2. J. Algebra 114 (1988), no. 1, 81–105.
Things can also be viewed in a very nice symmetric way using 4-ality. See Section 3.4 of 
J. Landsberg and L. Manivel, Representation theory and projective geometry.
In: Algebraic transformation groups and algebraic varieties, Enc.
Math. Sci., 132, Springer-Verlag, 2004, 131-167.