Quadratic forms permitting triple composition

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