Tensor algebras over the Steenrod algebra

It is known that unstable Steenrod module structure on the polynomial algebra $F_2[t_0,\dots ,t_{N-1}]\cong H^{⁎}({\left(R{P}^{\infty }\right)}^N;F_2)$ obtained by forgetting the multiplication is isomorphic to that arising from a twisted action of ${\mathrm{Sq}}^1$. We show that the same theorem holds for tensor algebras. As in the abelian case, the result is applied to produce a decomposition of the tensor algebra into “weight spaces”.