I know that for "regular" Whitehead product $[\cdot,\cdot]:\pi_k(X)\times\pi_l(X)\to\pi_{k+l-1}(X)$ there is Jacobi identity in the following form.
We have $\alpha\in \pi_k(X), ~\beta\in\pi_l(X), ~\gamma\in \pi_m(X)$ with $k,l,m>1.$ Then the following identity holds: $$(-1)^{km}[[\alpha,\beta],\gamma]+(-1)^{lk}[[\beta,\gamma],\alpha]+(-1)^{ml}[[\gamma,\alpha],\beta]=0.$$
I encountered that there exists the notion of a more general "universal" Whitehead product. For maps $f:\Sigma X\to Z$ and $g: \Sigma Y\to Z$ it is a map $[f,g]:\Sigma X\wedge Y\to Z$. Unfortunately I have not seen a lot of literature describing it and its properties. Some information is on pages 83-84 of Paul Selick's "Introduction to Homotopy Theory".
Is there a version of Jacobi identity for this Whitehead product? I would be glad if someone shared a reference to it and/or any discussion of this construction of Whitehead product.
Thank you.