  We prove that the quotients of the group algebra of the braid group
introduced by L. Funar in Comm. Math. Phys., 1995, collapses in characteristic
distinct from 2. In characteristic 2 we define several quotients of it, which
are connected to the classical Hecke and Birman-Wenzl-Murakami quotients, but
which admit in addition a symmetry of order 3. We also establish conditions on
the possible Markov traces factorizing through it.
