We extend the notion of the exterior Whitehead product for maps $$\alpha _i{:}\,\Sigma A_i \rightarrow X_i$$
for $$i=1,\ldots ,n$$
, where $$\Sigma A_i$$
is the reduced suspension of $$A_i$$
and then, for the interior product with $$X_i=J_{m_i}(X)$$
, the $$m_i$$
th-stage of the James construction J(X) as well. The main result stated in Theorem 4.10 generalizes (Hardie in Q J Math Oxford Ser 12(2):196–204, 1961, Theorem 1.10) and concerns to the Hopf invariant of the generalized Hopf construction. We close the paper applying Gray’s construction $$\circ $$
(called the Theriault product) to a sequence $$X_1,\ldots ,X_n$$
of simply connected co-H-spaces to obtain a higher Gray–Whitehead product map $$\begin{aligned} w_n{:}\,\Sigma ^{n-2}(X_1\circ \cdots \circ X_n)\rightarrow T_1(X_1,\ldots ,X_n), \end{aligned}$$
where $$T_1(X_1,\ldots ,X_n)$$
is the fat wedge of $$X_1,\ldots ,X_n$$
.