The main results of this paper are as follows. It is proved that in each \(\ell\)-group there exists a largest convex locally nilpotent \(\ell\)- subgroup. There is constructed a linearly orderable group G having a subgroup H of finite index with a nilpotent commutant such that the commutant of G fails to be nilpotent. A partial solution of question 12 in the problem list [Notices Am. Math. Soc. 29, 327 (1982)] is given.