The question of the existence of an analogue, in the framework of central simple algebras with involution, of the notion of Pfister form is raised. In particular, algebras with orthogonal involution which split as a tensor product of quaternion algebras with involution are studied. It is proven that, up to degree 16, over any extension over which the algebra splits, the involution is adjoint to a Pfister form. Moreover, cohomological invariants of those algebras with involution are discussed.

Abstract. For simple simply connected algebraic groups of classical type, Merkurjev, Parimala, and Tignol gave a formula for the restriction of the Rost invariant to the torsors induced from the center of the group. This paper completes their results by giving formulas for the exceptional groups. The method is somewhat different and also recovers their formula for classical groups.

Abstract. We use the fact that a projective half-spin representation of Spin 12 has an open orbit to generalize Pfister's result on quadratic forms of dimension 12 in I 3 to orthogonal involutions.In his seminal paper [Pf], Pfister proved strong theorems describing quadratic forms of even dimension ≤ 12 that have trivial discriminant and Clifford invariant, i.e., that are in I 3 . His results have been extended to quadratic forms of dimension 14 in I 3 by Rost, see [R] or [Ga]. One knows also extensions of these theorems where quadratic forms are replaced by central simple algebras with orthogonal involution, except in degree 12. The purpose of this paper is to complete this picture by giving the extension in the degree 12 case. The principle underlying the quadratic forms results and our extension is that a projective half-spin representation of Spin n for even n has an open orbit precisely for n ≤ 14, cf. [R], [I], and [SK].Let us first recall what is already known. We consider quadratic forms q of even dimension (resp. central simple algebras with orthogonal involution (A, σ) of even degree) with trivial discriminant and Clifford invariant. If q has dimension < 8, then q is hyperbolic by the Arason-Pfister Hauptsatz [L, X.5.1]. This also holds in the non split case: if A has degree < 8, then σ is hyperbolic, see e.g. [Ga 08, 1.4] or [Q, 4.4]. If q has dimension 8, then q is similar to a 3-Pfister form [L, X.5.6]; if A has degree 8, then (A, σ) is isomorphic to a tensor product ⊗ 3 i=1 (Q i , σ i ) of quaternion algebras with orthogonal involution [KMRT, 42.11]. If A has degree 10 or 14, then A is necessarily split [Ga 08, 1.5], so there is no interesting generalization of the theorem on quadratic forms. The remaining case is where q has dimension 12, where Pfister proved that q is isomorphic to φ ⊗ ψ for some 1-Pfister φ and 6-dimensional form ψ with trivial discriminant, see [Pf, pp. 123, 124] or [Ga, 17.13]. In Theorem 3.1 below, we prove an analogous statement for (A, σ) in case A has degree 12. We do not use Pfister's theorem on 12-dimensional quadratic forms in our proof, so we obtain his result as a corollary (Corollary 3.2).The paper comes in three sections. First, we study quadratic extensions of algebras with involution, and in particular orthogonal extensions of unitary involutions, as considered in [BP, App. 2], [ET, §3], and [QT, 2.14]. Second, we show how to construct an algebra with orthogonal involution that has trivial discriminant and Clifford invariant from any exponent 2 algebra with unitary involution. Third, we prove that in degree 12, this construction produces every central simple algebra with orthogonal involution that has trivial discriminant and Clifford invariant.In the language of linear algebraic groups, our results are as follows. Our construction takes a group of type 2

Abstract. We study central simple algebras with involution of the first kind that become hyperbolic over the function field of the conic associated to a given quaternion algebra Q. We classify these algebras in degree 4 and give an example of such a division algebra with orthogonal involution of degree 8 that does not contain (Q, ), even though it contains Q and is totally decomposable into a tensor product of quaternion algebras.