= Scalar extraction by iterated Weyl commutators
{title2=$(\operatorname{ad}D)^\alpha(\operatorname{ad}x)^\beta(a)=c_{\alpha\beta}\alpha!\beta!(-1)^{|\beta|}$}
For a nonzero element of a <Weyl algebra> over a characteristic-zero <field>, choose a monomial $x^\alpha D^\beta$ of maximal total degree with coefficient $c_{\alpha\beta}\ne0$. The displayed iterated <commutators> kill every lower-degree monomial. Another monomial of the same degree survives only if all its exponents dominate $(\alpha,\beta)$, which forces equality. Hence the result is a nonzero <scalar>. If the element has filtered degree at most $j$, expansion of these <commutators> gives a sum of terms $uav$ with both $u$ and $v$ of degree at most $j$. This proves <faithful finite-step action of a Weyl algebra> over any characteristic-zero <field>.
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