On the additive structure and asymptotics of codimensions $c_n$ in the algebra $F^{(5)}$

In this paper, we investigate the additive structure of the algebra $F^{(5)}$, i.e., a relatively free, associative, countably-generated algebra with the identity $[x_1, \dots, x_5] = 0$ over an infinite field of characteristic ${\neq}\, 2,3$. We study the space of proper multilinear polynomials in this algebra and means of basis construction in one of its basic subspaces. As an additional result, we obtain estimations of codimensions $c_n = \operatorname{dim} P_n / P_n \cap T^{(5)}$, where $P_n$ is the space of multilinear polynomials of degree $n$ in $F^{(5)}$ and $T^{(5)}$ is the $T$-ideal generated by the long commutator $[x_1, \dots, x_5]$.