Let $u_a(x),\,a=1,2,\ldots n$ be a $n$-component field in Minkowski spacetime $x^\mu,\,\mu=0,1,2,3$ and let $u_{a,\,\mu}=\frac{du_a}{dx^\mu}$. Let us introduce two operators (we use Einstein summation convention that repeated indices are implicitly summed over) $$\hat G=\tau^\mu\,\frac{d}{dx^\mu}+\sigma_a\,\frac{\partial}{\partial u_a}+ \sigma_{a,\mu}\,\frac{\partial}{\partial u_{a,\,\mu}}$$ and $$\hat N^\mu=\tau^\mu+\sigma_a\,\frac{\delta}{\delta u_{a,\,\mu}}+ \sigma_{a,\nu}\,\frac{\delta}{\delta u_{a,\,\mu\nu}},$$ where $$\sigma_a=\xi_a-\tau^\mu\,u_{a,\,\mu},$$ $\xi_a(x,u)$ and $\tau^\mu(x,u)$ do not depend on field derivatives, and $$\frac{\delta}{\delta u_{a,\,\mu}}=\frac{\partial}{\partial u_{a,\,\mu}}- \frac{d}{dx^\nu}\frac{\partial}{\partial u_{a,\,\mu\nu}},$$ $$\frac{\delta}{\delta u_{a,\,\mu\nu}}=\frac{\partial}{\partial u_{a,\,\mu\nu}}- \frac{d}{dx^\alpha}\frac{\partial}{\partial u_{a,\,\mu\nu\alpha}}.$$ It is claimed in http://link.springer.com/article/10.1023%2FA%3A1008240112483 (Lie–Bäcklund and Noether Symmetries with Applications, by N.H. Ibragimov, A.H. Kara and F.M. Mahomed) that $$[\hat G+\tau^\nu_{,\,\nu},\hat N^\mu]=\tau^\mu_{,\,\nu}\,\hat N^\nu. \tag{1}$$ The authors write that "the relation is proved by straightforward, albeit tedious, computation".

Is there a simpler way to prove the commutation relation (1)? I'm interesting to prove it only in the first jet space $(x^\mu,u_a,u_{a,\,\nu})$, so only first few terms from the cited article are presented in the definitions of $\hat G$ and $\hat N^\mu$.

Let $u_a(x),\,a=1,2,\ldots n$ be a $n$-component field in Minkowski spacetime $x^\mu,\,\mu=0,1,2,3$ and let $u_{a,\,\mu}=\frac{du_a}{dx^\mu}$. Let us introduce two operators (we use Einstein summation convention that repeated indices are implicitly summed over) $$\hat G=\tau^\mu\,\frac{d}{dx^\mu}+\sigma_a\,\frac{\partial}{\partial u_a}+ \sigma_{a,\mu}\,\frac{\partial}{\partial u_{a,\,\mu}}$$ and $$\hat N^\mu=\tau^\mu+\sigma_a\,\frac{\delta}{\delta u_{a,\,\mu}}+ \sigma_{a,\nu}\,\frac{\delta}{\delta u_{a,\,\mu\nu}},$$ where $$\sigma_a=\xi_a-\tau^\mu\,u_{a,\,\mu},$$ $\xi_a(x,u)$ and $\tau^\mu(x,u)$ do not depend on field derivatives, and $$\frac{\delta}{\delta u_{a,\,\mu}}=\frac{\partial}{\partial u_{a,\,\mu}}- \frac{d}{dx^\nu}\frac{\partial}{\partial u_{a,\,\mu\nu}},$$ $$\frac{\delta}{\delta u_{a,\,\mu\nu}}=\frac{\partial}{\partial u_{a,\,\mu\nu}}- \frac{d}{dx^\alpha}\frac{\partial}{\partial u_{a,\,\mu\nu\alpha}}.$$ It is claimed in http://link.springer.com/article/10.1023%2FA%3A1008240112483 (Lie–Bäcklund and Noether Symmetries with Applications, by N.H. Ibragimov, A.H. Kara and F.M. Mahomed) that $$[\hat G+\tau^\nu_{,\,\nu},\hat N^\mu]=\tau^\mu_{,\,\nu}\,\hat N^\nu. \tag{1}$$ The authors write that "the relation is proved by straightforward, albeit tedious, computation".

Is there a simpler way to prove the commutation relation (1)? I'm interested to prove it only in the first jet space $(x^\mu,u_a,u_{a,\,\nu})$, so only the first few terms from the cited article are presented in the definitions of $\hat G$ and $\hat N^\mu$.