On Fox spaces and Jacobi identities

In 1945, R. Fox introduced the so-called Fox torus homotopy groups in which the usual homotopy groups are embedded and their Whitehead products are expressed as commutators. A modern treatment of Fox torus homotopy groups and their generalization has been given and studied. In this note, we further explore these groups and their properties. We discuss co-multiplications on Fox spaces and a Jacobi identity for the generalized Whitehead products and the $\Gamma$-Whitehead products.