On $T$-spaces and relations in relatively free, Lie nilpotent, associative algebras

The purpose of this work is to obtain the commutator relations and Frobenius relations in a relatively free algebra $F^{(l)}$ specified by the identity $[x_1,\dots,x_l]=0$ over a field of characteristic $p>0$. These relations for $l>3$ are analogous to the relations in the algebra $F^{(3)}$ and are applied to the $T$-spaces in the algebra $F^{(l)}$. In order to study the relations in $F^{(l)}$ in more detail, we construct a model algebra analogous to the Grassmann algebra.