Let $𝔤$ be a complex, semisimple Lie algebra, with an involutive automorphism $\vartheta$ and set $𝔨=\mathrm{Ker}(\vartheta -I)$, $𝔭=\mathrm{Ker}(\vartheta +I)$. We consider the differential operators, $𝒟(𝔭{)}^K$, on $𝔭$ that are invariant under the action of the adjoint group $K$ of $𝔨$. Write $\tau :𝔨\to \mathrm{Der}𝒪\left(𝔭\right)$ for the differential of this action. Then we prove, for the class of symmetric pairs $(𝔤,𝔨)$ considered by Sekiguchi, that $\left\{d\in 𝒟\left(𝔭\right):d\left(𝒪(𝔭{)}^K\right)=0\right\}=𝒟\left(𝔭\right)\tau \left(𝔨\right)$. An immediate consequence of this equality is the following result of Sekiguchi: Let $({𝔤}_0,{𝔨}_0)$ be a real form of one of these symmetric pairs $(𝔤,𝔨)$, and suppose that $T$ is a $K_0$-invariant...