Supersymmetry algebras and Lorentz invariance for d=10 Super Yang-Mills

We consider ways in which conventional supersymmetry can be embedded in the set of more general fermionic transformations proposed recently [\Ref{B}] as a framework in which to study $d=10$ super Yang-Mills. Solutions are exhibited which involve closed algebras of various numbers of supersymmetries together with their invariance groups: nine supersymmetries with $\GT {\times}\SO (1,1)$ invariance; eight supersymmetries with $\SO (7){\times}\SO (1,1)$ invariance; four supersymmetries with $\SO (3,1){\times}\U (3)$ invariance. We recover in this manner all previously known ways of adding finite numbers of bosonic auxiliary fields so as to partially close the $d=10$ superalgebra. A crucial feature of these solutions is that the auxiliary fields transform non-trivially under the residual Lorentz symmetry, even though they are originally introduced as Lorentz scalars.