We study the recent construction of maximally supersymmetric field theory
Lagrangians in three spacetime dimensions that are based on algebras with a
triple product. Assuming that the algebra has a positive definite metric
compatible with the triple product, we prove that the only non-trivial examples
are either the well known case based on a four dimensional algebra or direct
sums thereof.Comment: 11 pages, very minor changes. Reference added. Version to be
published in JHE
