Poincar\'e superalgebras and triple systems

We consider a class of Poincar\'e superalgebras for which the nested bracket of three supercharges is necessarily zero only in dimensions greater than three. In lower dimensions, we give a precise characterisation of the data which encodes any such Poincar\'e superalgebra in terms of a more elementary embedding superalgebra. Up to isomorphism, we classify every classical embedding superalgebra that defines a Poincar\'e superalgebra. More generally, we show how to construct an embedding superalgebra in dimensions one, two and three from a certain type of triple system whose product structure encodes the nested bracket of three supercharges in the associated Poincar\'e superalgebra.