Abstract
In an algebraic investigation of isoparametric hypersurfaces, J. Dorfmeister and E. Neher encountered a nondegerate quadratic form which permitted composition with a trilinear product, Q ( { x y z } ) = Q ( x ) Q ( y ) Q ( z ) Q\left (\{{xyz} \} \right ) = Q(x)Q(y)Q(z) . In this paper we give a complete description of such composition triples: they are all obtained as isotopes of permutations of standard triples { x y z } = ( x y ) z \{xyz \} = (xy)z or x ( y z ) x(yz) determined by a composition algebra, with the quadratic form Q Q the usual norm form. For any fixed Q Q this leads to 1 1 isotopy class in dimensions 1 1 and 2 2 , 3 3 classes in the dimension 4 4 quaternion case, and 6 6 classes in the dimension 8 8 octonion case.
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