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A vector space $V$ with a symmetric trilinear map $V\times V \times V \to V$ will be called a triple system. Freudenthal introduced and studied a certain type of triple system on a 56-dimensional vector space whose automorphism group is a simply connected group of type E$_7$. If $S$ is a subsystem generated by $n$ generic elements in a triple system $V$, then (in some situations) we can use the slice method to deduce a surjection from $\mathrm{Aut}(V,S)$-torsors to $\mathrm{Aut}(V)$-torsors. In this talk I will describe some examples of this procedure for triple systems of varying dimensions; in one example we will deduce the upper bound on essential dimension $\mathrm{ed}(\mathrm{HSpin}_{12}) \leq 6$ for characteristic not 2 (beating the previously best known bound of 26 from Garibaldi and Guralnick).