much of the computational work in rough path theory relies on working with hall bases. You can find an interactive web page that generates hall bases for you at https://coropa.sourceforge.io/ Half way down the opening page. It is quite old but I think it still works. (the most up to date version of the code is in the source distribution on pypi for the package esig.

As others said, there is no unique hall basis.

much of the computational work in rough path theory relies on working with Hall bases. You can find an interactive web page that generates Hall bases for you at https://coropa.sourceforge.io/ Half way down the opening page. It is quite old but I think it still works. (the most up to date version of the code is in the source distribution on pypi for the package esig.

As others said, there is no unique hall basis.