I am a beginner, please help me to write a code to obtain the following formula, Thank you very much.
What have you tried?
I try to do check d^2=0 using the Mathematica, but I don't know how to write a code.
For what it's worth, I think you want to use (mathematical) wedge rather than tensor products here. You would actually need to define them first if you are not using any existing package that supports them.
There may be support already built into the add-on software xAct:
In particular you would want to check the xTerior sub-package.
If instead you want to implement something by hand, there are various threads on Mathematica.StackExchange.com about working with commutators. Also I have a section here called "Some noncommutative algebraic manipulation", that should give some ideas for how one could go about this.
Thank you very much for your comments! I will try my best to do it.