Orthogonal Involutions on Algebras of Degree 16 and the Killing Form of E 8

We exploit various inclusions of algebraic groups to give a new construction of groups of type E 8 , determine the Killing forms of the resulting E 8 's, and define an invariant of central simple algebras of degree 16 with orthogonal involution " in I 3 " , equivalently, groups of type D 8 with a half-spin representation defined over the base field. The determination of the Killing form is done by restricting the adjoint representation to various twisted forms of PGL 2 and requires very little computation. The first part of this paper (§ §1–6) extends the Arason invariant e 3 for quadratic forms in I 3 to central simple algebras (A, σ) " in I 3 " (this term is defined in §1) where A has degree 16 or has a hyperbolic involution. (The first case corresponds to simple linear algebraic groups of type D 8 with a half-spin representation defined over the base field.) The invariant e 3 detects whether (A, σ) is generically Pfister, see Cor. 2.6 below. We remark that the paper [BPQ] appears to rule out the existence of such an invariant by a counterexample. Our invariant exists exactly in the cases where their counterexample does not apply; surprisingly, this includes some interesting cases. The proofs in this part are not difficult, but we include this material to provide background and context for the later results. Proposition 1.4 generalizes the Arason-Pfister Hauptsatz for quadratic forms of dimension < 16, and depends on a result of Kirill Zainoulline presented in Appendix A. The real work begins in the second part of the paper (§ §7–10), where we use the inclusion PGL 2 × PSp 8 ⊂ PSO 8 to give a formula for the invariant in case (A, σ) can be written as a tensor product (Q,¯) ⊗ (C, γ), where (Q,¯) is a quaternion algebra with its canonical symplectic involution. We apply the preceding results in the third part of the paper (§ §11–16) to studying algebraic groups of type E 8. We give a construction of groups of type E 8 and compute the Rost invariant, Tits index (in some cases), and Killing form of the resulting E 8 's, see Th. 9.1, Prop. 12.1, and Th. 15.2. We compute the Killing form by branching to subgroups of type A 1 , which is somewhat cleaner than computations of other Killing forms …