  Given a quasi-reductive algebraic supergroup $G$, we use the theory of
semisimplifications of symmetric monoidal categories to define a symmetric
monoidal functor $\Phi_x: \Rep(G) \to \Rep(OSp(1|2))$ associated to any given
element $x \in \mathrm{Lie}(G)_{\bar 1}$. For nilpotent elements $x$, we show
that the functor $\Phi_x$ can be defined using the Deligne filtration
associated to $x$.
  We use this approach to prove an analogue of the Jacobson-Morozov Lemma for
algebraic supergroups. Namely, we give a necessary and sufficient condition on
odd nilpotent elements $x\in \mathrm{Lie}(G)_{\bar 1}$ which define an
embedding of supergroups $OSp(1|2)\to G$ so that $x$ lies in the image of the
corresponding Lie algebra homomorphism.
