You should be interested in the discussion in 

<cite authors="Evans, Gareth A.; Wensley, Christopher D.">_Evans, Gareth A.; Wensley, Christopher D._, [**Complete involutive rewriting systems**](https://doi.org/10.1016/j.jsc.2007.07.005), J. Symb. Comput. 42, No. 11-12, 1034-1051 (2007). [ZBL1128.68044](https://zbmath.org/?q=an:1128.68044).</cite>

on the notion of noncommutative involutive systems, which in the commutative case are a well known modification of the Gröbner basis theory. Evans' thesis is available at [math.RA/0602140](https://arxiv.org/abs/math/0602140 "Noncommutative Involutive Bases"). 

I can't help on the proposed applications. 

Ronnie Brown