Minuscule embeddings

We study embeddings J→G of simple linear algebraic groups with the following property: the simple components of the J module Lie(G)/Lie(J) are all minuscule representations of J. One family of examples occurs when the group G has roots of two different lengths and J is the subgroup generated by the long roots. We classify all such embeddings when J=SL2 and J=SL3, show how each embedding implies the existence of exceptional algebraic structures on the graded components of Lie(G), and relate properties of those structures to the existence of various twisted forms of G with certain relative root systems.