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?)

Lie Algebras of Finite and Affine Type(Cambridge, 2005), including the choice of basis under a graph automorphism. Carter's book is also useful because he later generalizes many of the ideas to affine Lie algebras. $\endgroup$