#### Vol. 304, No. 1, 2020

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Decomposability of orthogonal involutions in degree $12$

### Anne Quéguiner-Mathieu and Jean-Pierre Tignol

Vol. 304 (2020), No. 1, 169–180
##### 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. Our main result 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 $2$.

##### Keywords
algebra with involution, hermitian form, cohomological invariant
##### Mathematical Subject Classification 2010
Primary: 11E72
Secondary: 11E81, 16W10