  An off-shell formulation for 6 and 10 dimensions simple supersymmetric
Yang-Mills theories is presented. While the fermionic fields couple to left
action of S^3 and S^7 respectively, the auxiliary ones couple to right action
(and vice versa). To close the algebra off-shell, left and right actions must
commute. For 6 dimensions quaternions work fine. The 10 dimensional case needs
special care. Pure spinors and soft Lie algebra (algebra with structure
functions instead of structure constants) are essential. Some tools useful for
constructing the superspace are also derived. We show how to relate our results
to the early works of Evans and Berkovits.
