There are lots of papers dealing with representation-theoretic questions and universal enveloping algebras using Gröbner bases. Some examples are given by these: [1][1], [2][2], [3][3], [4][4].

  [1]: https://s-space.snu.ac.kr/bitstream/10371/12179/1/42.pdf "Kang, Seok-Jin; Lee, Kyu-Hwan, Gröbner-Shirshov bases for representation theory, J. Korean Math. Soc. 37, No. 1, 55-72 (2000). Available at https://jkms.kms.or.kr/journal/view.html?uid=1005. zbMATH review at https://zbmath.org/0979.16010"
  [2]: https://doi.org/10.1090/S1061-0022-2011-01159-8 "Koryukin, A. N., Gröbner-Shirshov bases of the Lie algebra D_n^+, St. Petersbg. Math. J. 22, No. 4, 573-614 (2011); translation from Algebra Anal. 22, No. 4, 76-136 (2010). zbMATH review at https://zbmath.org/1244.17008"
  [3]: http://www.springerlink.com/content/bwn7336156456u7u/
  [4]: https://doi.org/10.1142/S0218196796000234 "Bokut, L. A.; Klein, A. A., Serre relations and Gröbner-Shirshov bases for simple Lie algebras. II, Int. J. Algebra Comput. 6, No. 4, 401-412 (1996). zbMATH review at https://zbmath.org/0866.17008"