Construction and applications of an additive basis for the relatively free associative algebra with the lie nilpotency identity of degree 5

We construct an additive basis for the relatively free associative algebra $F^{(5)}(K)$ with the Lie nilpotency identity of degree 5 over an infinite domain $K$ containing $\tfrac{1}{6}$. We prove that approximately half of the elements in $F^{(5)}(K)$ are central. We also prove that the additive group of $F^{(5)}(\Bbb Z)$ lacks the elements of simple degree $\ge 5$. We find an asymptotic estimation of the codimension of T-ideal, which is generated by the commutator $[x_1, x_2,\dots,x_5 ]$ of degree 5.