Osp(1|2) and Sl(2) Reductions in Generalised Super-Toda Models and Factorization of Spin 1/2 Fields

I show that the classical Toda models built on superalgebras, and obtained from a reduction with respect to an $Sl(2)$ algebra, are "linearly supersymmetrizable" (by adding spin 1/2 fields) if and only if the $Sl(2)$ is the bosonic part of an $OSp(1|2)$ algebra. In that case, the model is equivalent to the one constructed from a reduction with respect to the $OSp(1|2)$ algebra, up to spin 1/2 fields. The corresponding $W$ algebras are related through a factorization of spin 1/2 fields (bosons and fermions). I illustrate this factorization on an example: the superconformal algebra built on $Sl(n+1|2)$.