A Remark on the Realization of Mappings of the 3-Dimensional Sphere into Itself

The problem of realizing a mapping $f\colon S^3 \to S^3$ of the $3$-dimensional sphere into itself in the ambient space $\mathbb R^6$ is reformulated in elementary terms. It is proved that, for $n=1,3,7$, there exists an equivariant mapping $F\colon S^n\times S^n\to S^n\times S^n$ such that a formal obstruction to its realization in $\mathbb R^{2n}$ is nontrivial.