Decomposability of orthogonal involutions in degree 12

Abstract

A theorem of Pfister asserts that every 12-dimensional quadratic form with trivial discriminant and trivial Clifford invariant over a field of characteristic different from 2 decomposes as a tensor product of a binary quadratic form and a 6-dimensional quadratic form with trivial discriminant. The main result of the paper extends Pfister's result to orthogonal involutions : every central simple algebra of degree 12 with orthogonal involution of trivial discriminant and trivial Clifford invariant decomposes into a tensor product of a quaternion algebra and a central simple algebra of degree 6 with orthogonal involutions. This decomposition is used to establish a criterion for the existence of orthogonal involutions with trivial invariants on algebras of degree 12, and to calculate the f 3 -invariant of the involution if the algebra has index 2.

Minuscule embeddings

“…Every inner form H of H 0 is isogenous to SO(A, σ) where A is a central simple k-algebra of dimension 12 2 and σ is an orthogonal involution with trivial discriminant such that the even Clifford algebra C(A, σ) as defined in [8, §8] has one split component (namely the action on W). Such pairs (A, σ) have recently been described more explicitly, see [29]. Such an H is isotropic if and only if the involution σ is isotropic, i.e., if and only if σ(a)a = 0 for some nonzero a ∈ A.…”