Seminar at SMU Delhi

Let $k$ be a field of characteristic different from $2$ and $3$. Let $G$ be a simple algebraic group of type $G_2$ or $F_4$ over $k$. It is known that groups of type $G_2$ over $k$ occur as groups of automorphisms of octonion algebras and groups of type $F_4$ over $k$ are precisely the groups of automorphisms of Albert algebras. These algebras define certain mod-2 (Galois cohomological) invariants of these groups. In this talk we will discuss how these invariants determine $k$-embeddings of groups of type $A_1$ and $A_2$ in $G$. Moreover, $k$-groups of type $F_4$ (resp. $G_2$) are generated by $k$-subgroups of type $A_2$ (resp. $A_1$). This motivates the study of embeddings mentioned above.