The Operator Algebra of Hypersurface Deformation Operators

Question

Let $\Sigma$ and ${M}$ be smooth manifolds with $\mathrm{dim}(\Sigma)<\mathrm{dim}({M})$, as well as the coordinate charts $\xi$ on $\Sigma$ and ${x}$ on ${M}$. Let further be $\Phi\in{C}^{\infty}(\Sigma,{M})$ a smooth embedding map which induces a hypersurface $\Phi(\Sigma)$ in ${M}$. Let ${N}\in\Gamma^{\infty}({M},{T}^{\star}{M})$ with ${0}=(\Phi^{\star}{N})_{\alpha}(\xi)=\frac{\partial\Phi^{\mu}(\xi)}{\partial\xi^{\alpha}}{N}_{\mu}({x}(\xi))$ be a covector field that annihilates any vectorfield in $\Gamma^{\infty}({M},{T}{M})$ that is tangent to $\Phi(\Sigma)$, with the notation being $\Phi^{\mu}(\xi)={x}^{\mu}(\Phi(\xi))$. In order to understand how functionals of the embedding map ${F}[\Phi]$ change under a deformation of the hypersurface $\Phi(\Sigma)$, given through the vectorfield ${A}\in\Gamma^{\infty}({M},{T}{M})$, one needs to decompose the total hypersurface deformation into hypersurface normal and hypersurface tangent directions respectively, i.e. ${A}^{\mu}(\Phi(\xi))={X}(\xi){N}^{\mu}(\Phi(\xi))+{X}^{\alpha}(\xi)\frac{\partial\Phi^{\mu}(\xi)}{\partial\xi^{\alpha}}$, where ${X}\in{C}^{\infty}(\Sigma)$ and ${X}^{\alpha}\frac{\partial}{\partial\xi^{\alpha}}\in\Gamma^{\infty}(\Sigma,{T}\Sigma)$. The hypersurface normal deformation operator is given through \begin{align*} \mathcal{H}({X})=\int_{\Sigma}\mathrm{d}\xi{\,}{X}(\xi){g}^{\mu\nu}(\Phi(\xi)){N}_{\mu}(\Phi(\xi))\frac{\delta}{\delta\Phi^{\nu}(\xi)}{\,}{.} \end{align*} The hypersurface tangent deformation operator is given through \begin{align*} \mathcal{D}({X}^{\alpha}\partial/\partial\xi^{\alpha})=\int_{\Sigma}\mathrm{d}\xi{\,}{X}^{\alpha}(\xi)\frac{\partial\Phi^{\mu}(\xi)}{\partial\xi^{\alpha}}\frac{\delta}{\delta\Phi^{\mu}(\xi)}{\,}{.} \end{align*} The algebra of these operators is given through \begin{align*} [\mathcal{H}({X}),\mathcal{H}({Y})]&=-\mathcal{D}((\Phi^{\star}{g})^{\alpha\beta}({Y}\partial{X}/\partial\xi^{\alpha}-{X}\partial{Y}/\partial\xi^{\alpha})\partial/\partial\xi^{\beta}){\,}{,}\\[0.5em] [\mathcal{D}({X}^{\alpha}\partial/\partial\xi^{\alpha}),\mathcal{H}({Y})]&=-\mathcal{H}({X}^{\alpha}\partial{Y}/\partial\xi^{\alpha}){\,}{,}\\[0.5em] [\mathcal{D}({X}^{\alpha}\partial/\partial\xi^{\alpha}),\mathcal{D}({Y}^{\beta}\partial/\partial\xi^{\beta})]&=-\mathcal{D}(({X}^{\alpha}\partial{Y}^{\beta}/\partial\xi^{\alpha}-{Y}^{\alpha}\partial{X}^{\beta}/\partial\xi^{\alpha})\partial/\partial\xi^{\beta}){\,}{.} \end{align*} But when I try to derive this algebra from the definitions above I get \begin{align*} \mathcal{D}({X}^{\alpha}\partial/\partial\xi^{\alpha})\mathcal{D}({Y}^{\beta}\partial/\partial\xi^{\beta}){F}[\Phi]&=\int_{\Sigma\times\Sigma}\mathrm{d}\xi{\,}\mathrm{d}\xi^{\prime}{\,}{X}^{\alpha}(\xi)\frac{\partial\Phi^{\mu}(\xi)}{\partial\xi^{\alpha}}\frac{\delta}{\delta\Phi^{\mu}(\xi)}\bigg({Y}^{\beta}(\xi^{\prime})\frac{\partial\Phi^{\nu}(\xi^{\prime})}{\partial\xi^{\prime\beta}}\frac{\delta{F}[\Phi]}{\delta\Phi^{\nu}(\xi^{\prime})}\bigg)\\[0.5em] &=\int_{\Sigma\times\Sigma}\mathrm{d}\xi{\,}\mathrm{d}\xi^{\prime}{\,}{X}^{\alpha}(\xi){Y}^{\beta}(\xi^{\prime})\bigg(\frac{\partial\Phi^{\mu}(\xi)}{\partial\xi^{\alpha}}\frac{\partial\delta(\xi-\xi^{\prime})}{\partial\xi^{\prime\beta}}\frac{\delta{F}[\Phi]}{\delta\Phi^{\mu}(\xi^{\prime})}+\frac{\partial\Phi^{\mu}(\xi)}{\partial\xi^{\alpha}}\frac{\partial\Phi^{\nu}(\xi^{\prime})}{\partial\xi^{\prime\beta}}\frac{\delta^{2}{F}[\Phi]}{\delta\Phi^{\mu}(\xi)\delta\Phi^{\nu}(\xi^{\prime})}\bigg){\,}{,} \end{align*} \begin{align*} [\mathcal{D}({X}^{\alpha}\partial/\partial\xi^{\alpha}),\mathcal{D}({Y}^{\beta}\partial/\partial\xi^{\beta})]{F}[\Phi]&=\int_{\Sigma\times\Sigma}\mathrm{d}\xi{\,}\mathrm{d}\xi^{\prime}{\,}{X}^{\alpha}(\xi){Y}^{\beta}(\xi^{\prime})\bigg(\frac{\partial\Phi^{\mu}(\xi)}{\partial\xi^{\alpha}}\frac{\partial\delta(\xi-\xi^{\prime})}{\partial\xi^{\prime\beta}}\frac{\delta{F}[\Phi]}{\delta\Phi^{\mu}(\xi^{\prime})}-\frac{\partial\Phi^{\mu}(\xi^{\prime})}{\partial\xi^{\prime\beta}}\frac{\partial\delta(\xi-\xi^{\prime})}{\partial\xi^{\alpha}}\frac{\delta{F}[\Phi]}{\delta\Phi^{\mu}(\xi)}\bigg)\\[0.5em] &=-\int_{\Sigma\times\Sigma}\mathrm{d}\xi{\,}\mathrm{d}\xi^{\prime}{\,}\delta(\xi-\xi^{\prime})\bigg({X}^{\alpha}(\xi)\frac{\partial\Phi^{\mu}(\xi)}{\partial\xi^{\alpha}}\frac{\partial}{\partial\xi^{\prime\beta}}\bigg({Y}^{\beta}(\xi^{\prime})\frac{\delta{F}[\Phi]}{\delta\Phi^{\mu}(\xi^{\prime})}\bigg)-{Y}^{\beta}(\xi^{\prime})\frac{\partial\Phi^{\mu}(\xi^{\prime})}{\partial\xi^{\prime\beta}}\frac{\partial}{\partial\xi^{\alpha}}\bigg({X}^{\alpha}(\xi)\frac{\delta{F}[\Phi]}{\delta\Phi^{\mu}(\xi)}\bigg)\bigg){\,}{.} \end{align*} Does anyone know where my mistake lies and how to do it for the other two commutation relations? Thanks for any help in advance.