Invariants mod-2 and subgroups of [formula omitted] and [formula omitted

Abstract

Let k be a field of characteristic different from 2 and 3. Let G be a simple group of type F4 or G2 defined over k. In this paper we discuss embeddings of connected simple algebraic groups of type A1 and A2 in G in terms of the mod-2 Galois cohomological invariants attached to these groups. We prove that k-groups of type F4 (resp. G2) arising from division algebras are generated by k-subgroups of type A2 (resp. A1) (see Theorems 3.11, 4.1). We derive a necessary and sufficient condition for an Albert algebra to have zero f5 invariant (Theorem 3.4). Further, for a k-group G of type F4, we derive a condition necessary for a k-group H of type A1 or A2 to embed in G over k. We prove that in order to embed H in G over k, the mod-2 invariant of H must divide f5(G) (see Section 2 and Remarks 2.8, 2.3 for definition of invariants). Along similar lines, we derive a condition for a k-group H of type A2 to embed in a k-group G of type G2. We prove that H embeds in G over k, if and only if f3(H)=f3(G) (Theorem 4.4). Next we derive a necessary and sufficient condition for a k-group H of type A1 to embed in a k-group G of type G2. If G=Aut(C) is a group of type G2 over k for an octonion algebra C over k, then this condition provides a natural bijection between k-conjugacy classes of involutions in G(k) and isometry classes of 2-fold Pfister divisors of the Pfister form nC, the norm form of the octonion algebra C (Section 4, Proposition 4.2).