It’s maybe impressive because it’s very compact as a computation but when you know the definition of the Lie bracket i.e. [u , v] = uv – vu and that it is antisymmetric i.e. [u,v]=-[v,u], you understand that the first step is using this property, that the second is the application of the definition, the third its expansion, the fourth its rearrangement etc.

BTW there is an easier to solve this exercise which is to identify the total expansion of the two members of the equality but it would less impressive and less elegant. 

It’s a way to prove the beautiful Jacobi relation i.e. [x,[y,z]] + [z,[x,y]] + [y,[z,x]]=0 Where we see the nice cyclic permutation : x,y,z & z,x,y & y,z,x. 

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