# transformations with commutators and anticommutators that generate displacements

is well known that composition of point reflections generate pure displacements. This implies that the commutator of two point reflections will be a pure displacement. Are there similar elemental transformation operations which can generate pure displacements from their anticommutators?

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$$PQf(x) = f(x-(p-q)) \\ QPf(x) = f(x-(q-p))$$ $$[P,Q]f(x) =(PQ-QP)f(x) = f(x-p+q) - f(x-q+p)$$
If you take, for example $f(x) = x$, then $[P,Q]x = x-p+q-x+q-p = 2(q-p)$. This constant is clearly not a translation of $x$.